The STG44 was one of the first commonly issued weapons using an what we would call an intermediate round, bridging the gap between conventional rifle calibers, and the smaller pistol rounds commonly used in submachine guns.
In so many words it was a weapon that retained a lot of the ranged utility of a regular rifle/carbine while still having a lot more utility at close quarters.
The BAR and FG42 both used full sized rifle cartridges and had several characteristics that made them not assault rifles:
The M1918 was always intended as a light machine gun sort of weapons system. It was designed at a time in which you basically had either heavy belt fed machine guns, or smaller machine guns fed from magazines. It was quite large, and while still fairly maneuverable for it's weight/weapons type, still served effectively as a support by fire vs assault weapon.
The FG42 is an odder duck simply because of how few were made and used (basically it was used as a bolt action rifle replacement, serving alongside troops that still had K98s, MP40's and MG42s vs being forced into the support or close in weapons use it was supposed to also do). It was intended be a bit of a "do-all" weapons system replacing SMGs, bolt action rifles, and by some accounts, some of the MGs used by German paratroopers. In practice however, while it was fairly light, the full sized rifle cartridges tended to make automatic fire unmanageable, meaning in practice you could argue it was closer to what is today referred to as a "battle rifle" (M14, FN FAL, G3 being good analogs).
A good also-ran contender for first assault rifle would be the M1/M2 Carbine, given it too used a sort of intermediate round (although leaning more towards big pistol than small rifle), with the later M2 offering fully automatic fire and a large magazine.
The understanding of what makes an assault rifle is about as precise- especially in that early 20th century era of fire arms- is about as specific as Doctor Who's definition of time.
The Browning Automatic Rifle is more of a crazy uncle to the assault rifle. Indeed, it was a rifle, but it fired full rifle cartridges, which are considered too large for an assault rifle, which specifically used rounds that were sized larger than pistol rounds, but smaller than full rifle rounds. Furthermore, while the BAR did have a fire selector, the choices were between automatic fire, and faster automatic fire.
The FG42 also wasn't an assault rifle for a similar reason; it fired a full rifle round. In a softer sense, it's really hard to argue that a rifle that is prone to failure from aggressive handling is going to be viable as a rough and tumble assault rifle. These days common tests to 'punish' an assault rifle to see if it is combat worthy involve letting it bathe in water, crud, and mud and then immediately pulling it out to see if it will fire.
The generally agreed on definition of an assault rifle is a gas powered rifle that fires an intermediate cartridge, fed from a detatchable magazine, with a fire select that allows for automatic, semiautomatic and (perhaps) burst fire. Many rifles from the early 1900's through WW2 were often only one feature short of being considered an assault rifle. The Fedorov Avtomat is essentially an assault rifle. Although it does fire a full rifle round, the Arisaka 6.5x50 is small for it's class. Similarly the Italians had the Cei-Rigotti and while the versions of it that saw use used internal magazines, there were (allegedly) variants that were in the prototyping stage that had detachable magazines. The British also had the Lewis Gun and the Canadians cooked up the Huot Automatic Rifle both of which also flirt with being a full on assault rifle.
Part of the problem is that many guns that are technically very close to being assault rifles were being marketed to a world that wanted something akin to a light machine gun. As a result more than one 'uncle' to the assault rifle spent it's life being considered a light machine gun- the Lewis gun and the Fedorov for example. There was little concept of an assault rifle at this stage in history and even as it emerged the logistical capacity of a standing, active army numbering in the millions is not going to be able to rapidly retool itself from supply trains to manufacturing centers to stop making one form of rifle and ammunition and start using another. Even if a rifle is adopted by a country's army you're looking at easily a decade worth of teething the design and working out flaws. The M1 Garand was first tested in the early 1920's but would not hit production till 1936, and redesigns wouldn't stop till 1940ish.
Even the most revolutionary rifle design is fundamentally hamstrung by the fact that no matter how accurate, how quickly it fires, how durable, reliable, and how cheap it is to make, it has to grapple with the fact that two legs move it around and two hands have to aim it. And off the front you're looking at feasibly tens and hundreds of boots on the ground operating factories to produce the guns, repair kits, spare parts, and munitions. Or to put it in other terms, the earliest grandfather of the machine gun (or at least the Gatling gun) was patented in 1718. Just because an idea is great doesn't mean it'll catch on, or even that you necessarily have the technology to truly bring it to life.
Defining what an assault rifle is can be a bit of contentious issue, but there reasons why by the FG-42 and BAR are not classified as such: What makes something "not an assault rifles" is simple in theory (it boils down really to what cartridge they fire, among other things) but the semantics can be difficult to discern. So lets look at what we define an assault rifle as being and the genesis of the concept, then look at each weapon and look at what makes them "not an assault rifle".
What makes the STG 44 the first assault rifle?
In simple terms: firing an intermediate cartridge rather than a full size rifle cartridge, and select fire capability. The round fired by the STG 44 was the 7.92×33mm Kurz cartridge, basically the standard German 7.92x57mm Mauser rifle cartridge adapted by shortening the case (Kurz=short in German). The point of this development was the capability gap between the two main German infantry small arms, the Kar 98K bolt action rifle, and the MP series sub-machine gun. The Kar 98 was an accurate weapon, effective over long range, and the full size Mauser cartridge had good killing power, but the small magazine capacity, low rate of fire and large length and weight of the weapon proved disadvantageous in short range and urban engagements. In comparison, the MP38/MP40 sub-machine gun, firing the 9mm Parabellum pistol cartridge was an excellent weapon in close quarters fighting, by dint of its compact size and weight, automatic fire, and 32 round magazine. Conversely, the pistol caliber cartridge and short barrel left the MP 40 at a loss in longer range engagements.
Thus is the genesis behind the 7.92×33mm Kurz cartridge and STG 44: a weapon providing an acceptable trade-off between the strong points of both weapons, firing a cartridge that lies between the full size rifle cartridge and pistol caliber cartridge in terms of both size and performance. The outcome was a both a weapon could replace both roles fairly satisfactorily, as well as a massive increase in the firepower available to the German infantry squad; smaller, lighter rounds meant more ammunition could be carried, more rounds could be fit into the detachable magazines, and that the weapon was fairly controllable when fired on full auto. The significance was a vast improvement on the average fire rate of an infantry squad, as previously, German infantry tactics were centered around the medium machine gun as the linchpin of a squad's firepower and ability to fire and maneuver. Prior to the STG 44, battle experience showed that a weakness of these tactics was that the weight of fire and ability to maneuver was massively reduced when the machine gun was not firing, due to having to reload, change barrels or maneuver into a new position. The STG 44 not only improved the overall firepower available to an infantry squad, but also improved the average fire rate over the length of a typical infantry engagement.
Why aren't these other weapons assault rifles then?
We can discount the BAR as an assault rifle fairly easily: The M1918 Browning Automatic Rifle was designed on the principle of providing a automatic infantry weapon for use in the "walking fire" role in WW1; walking fire was an envisioned assault tactic which involved infantry sections attacking enemy positions under the cover of a hail of lead, provided by a man-portable automatic weapon fired from the hip while walking by 1 designated user. It also was to act as the overall replacement in US service to the various different light machine guns the US had adopted, like French Chauchat light machine gun, another "walking fire" weapon which had proven deeply unpopular in US service. Although we could argue that the BAR is indeed a weapon for assaulting things, it isn't an "assault rifle" under our definition for good reason. Firing a full size rifle cartridge is one disqualifying design point, the other is role: Whereas the STG 44 is a trade-off between the rifle and the sub-machine gun, the BAR is a different trade-off made between the bolt action infantry rifle and the heavy, emplaced belt-fed machine gun. Arguably, the BAR is not a particularly good trade-off; although possessing a definite firepower advantage over than bolt action rifle, the BAR was heavy and awkward, and pretty poor in the sustained fire role thanks to its disappointing 20-round box magazine. Whereas the STG 44 was meant for issue to all the riflemen of a squad, the M1918 was only meant for issue to one per squad, which is why we file it under the light machine gun family of weapons.
As for the FG 42, it is a harder case to argue, but I still am convinced it is not a "pure" assault rifle. The oddly futuristic looking FG-42 was a response to the need to increase the firepower available to German paratroopers (Fallschirmjager) by giving them a select-fire weapon small enough to parachute with, as a common complaint from Fallschirmjager operations was that they were left badly outgunned in early stages of airborne operations, due to the fact that they dropped with their small arms and support weapons in crates, due to a shortcoming of the German parachute harness that meant large weights such as rifles and machine guns could not be jumped with.
Does this make it an assault rifle? It's a tricky question to answer. Whereas the STG 44 was meant as a replacement for the sub machine gun and rifle in a squad, the FG 42 was a phenomenally advanced weapon, designed to go even further by aiming to cover the role of rifle, machine gun and sub-machine gun in a paratrooper section. This, combined with the the fact that it fires the full sized 7.92mm rifle cartridge, is what disqualifies it in my eyes: Rather than sharing the same mantle as the STG 44, the FG 42 is instead a unique select fire rifle, designed for a relatively small production run in a very niche sphere. In some peoples eyes the FG 42 is grouped under the light machine gun family, but I feel this is an oversimplification; it borrows elements from all the small arms of a paratrooper section, but arguably most heavily from the rifle and the support machine gun.
An assault rifle is, generally, a select fire rifle that uses large detachable box magazines and an intermediate rifle round (lower power than the traditional pre-WWII military rifle rounds like .30-06 and similar). The advantage of such a weapon is that it straddles the role between a rifle and a sub-machine gun. The experience of WWII and later wars involving major industrialized armies was that engagements were typically at shorter distances (a few hundred yards at most versus many hundred yards) than theory/doctrine expected, and that the way to win battles was to use volume of fire to control the enemy's movements so you could maneuver against them (versus a few very targeted shots all designed to kill a specific enemy).
Assault rifles did a better job serving the actual needs of battle than sub-machine guns, service rifles, or the combination of both. The select fire (automatic fire option) meant that it was easier to deliver a large volume of fire to suppress the enemy at typical engagement ranges. The intermediate caliber rounds were effective at the actual engagement distances encountered but were lighter so soldiers could carry more (and were more effective beyond point-blank range than pistol rounds fired but SMGs). The lower recoil from the intermediate rounds meant that the rifles were more controllable and easier to actually hit a target with. The smaller rounds meant the whole rifle could be built lighter, making it easier for one person to carry throughout a war on foot.
The high capacity detachable box magazines meant that a soldier could maintain fire on a target for longer than older rifle magazines, and could be reloaded very rapidly. The M1 Garand has an 8 round en bloc clip magazine, the K98k has a 5 round internal magazine fed by stripper clips. To fire 50 rounds at an enemy someone with a Garand needs to reload 6 times, each time taking their sights off the target for from 5 to maybe 20 seconds, depending on how fast they are at finding their fresh clips and at reloading (for a half a minute to 2 whole minutes spent reloading). To fire 50 rounds with a K98k someone would have to reload 10 times, taking maybe 5 to 20 seconds doing so, for a total of around 1-3 minutes spent reloading. To fire 50 rounds from an Stg-44 requires one reload, for a total of about 5 seconds spent reloading, and they still have 10 rounds remaining in their magazine.
The BAR doesn't fit the definition because it used a full-size rifle round. It was big, it was heavy (50% heavier than the Stg-44, unloaded), the ammo was big and heavy. It had a ton of recoil, making it very challenging to fire accurately at full auto. For these reasons the BAR wasn't issued as a standard weapon for entire squads, it was used as a light machine gun or special weapon serviced by a team of 2 or 3 soldiers. In the few cases where a BAR was used by one-man teams the ammunition was still distributed among other soldiers in the squad. The ammo was just too heavy to carry a reasonable amount of it on one person. In short, the BAR wasn't suitable as a complete replacement for the standard service rifle in units, and it couldn't reasonably be issued to everyone as their main weapon. To put things in perspective, someone carrying a BAR with only 40 rounds (two magazines) worth of weight is the equivalent of someone carrying an Stg-44 and around 200 rounds of ammo (neglecting magazine weight).
What were the structural forces that allowed this ballistic and weapon come through?
I assume a longer pointier bullet would have been known at the time to be more accurate. Was a change of weapon doctrine instrumental? Or was it purely advances in technology that allowed for an already sought goal (to produce an assault rifle) to be attained?