Well, first, let's explain something to start: what you are referring to are called casemate tank destroyers, many of which are also called assault guns. There are also lightly armoured tank destroyers that have partial turrets, a design more popular with the Americans in the war.
As to why they didn't remain relevant after the war; well, that begs the question: what drove German and Soviet designers to make these AFVs in the first place? I mentioned above that these weapons platforms were also often called assault guns. Most of the most well known tank destroyers for both sides started out primarily as these types of weapons. The Stug III was designed primarily as a weapon to support infantry assaults on fortified targets, hence the large frontal armour and the perceived lack of a need of a turret. When your target is a stationary fortress, you don't need the later, but you definitely need the former. In that vein, it was armed with the 7.5 cm KwK 37 L/24, a low-velocity gun designed primarily to be used with HE rounds in a direct fire role, not against armoured targets.
Fast forward to 1941 and the Germans are invading the USSR and finding that their inventory of tanks are struggling against the KVs and T-34s of the Red Army. A rapid answer to this needs to be found and fielded. The Panzer III was designed to be the tank for the anti-tank role in the Wehrmacht, but it's turret was simply too small to handle a larger weapon with retain stability. The solution was to replace the aformentioned support weapon in the Panzer IV and Stug IIIs with the 7.5 cm KwK 40. This was a much longer barrel, and actually barely fit with the Panzer IV, while the Stug was much more easily modified to fit this new weapon. And, ultimately, this is where we come to the point of our little story. You see, turrets and tanks themselves can only handle a certain size increases from their initial armament before the entire platform just doesn't work anymore. You can't take a turret designed for a 30mm gun and just shove a 75mm gun in there, or a turret designed for a 50mm gun and shove an 88mm gun in there. Larger guns also can tax existing other systems in the tank, and may also seriously jeopardize balance on the move (one of the reasons the Panzer IV's 7.5 cm KwK 40 was shortened). And, perhaps most importantly, the suspension and engine can only handle a certain amount of weight, and a larger gun and possibly a larger turret can only go so far with an existing suspension and powerplant.
However, with these assault guns/TDs, that wasn't a problem. You didn't need to worry about a turret, so a lot of weight and space was saved. This meant that existing platforms could be re-purposed. If you had a platform with a suspension and powerplant that could handle 20 tons, eliminating a turret means that weight can be transfered to a larger weapon and more armour. The Stug III was based on the Panzer III, that tank mentioned earlier that was not up-gradable. Well, that platform was very well understood, reliable, and tooling was already available for mass production. We see similar reasons for the increase in production of the Jagdpanzer IV later in the war as the Panzer IV saw it's usefulness as a front line tank deminish, the tooling was reused.
So, why weren't they used after the war? Well, not having a turret represents a pretty huge compromise. These AFVs were best utilized in their anti-armour roles when they could lay in ambush, with well protected flanks and easily lines of retreat. They were not very useful for attacking. However, in the quick pace of armoured development in WW2, they represented a stop-gap for both sides to deal with armoured threats as existing turreted platforms were replaced or upgraded. A great example of this is the SU-85 for the Red Army. A relatively well known TD, it existed to counter threats such as the new Panther and other heavier tanks for the Germans. However, as soon as the T-34 was redesigned with a larger turret to fit the same 85mm cannon, production was stopped.
See, a tank with a turret is almost always preferable to one without. Except in one particular case. Remember those Stugs with the low-velocity gun? An AFV designed for frontal assaults on stationary targets still doesn't need a turret, and indeed, several of these designs continued to be used during and after the war without any modifications made for TD purposes. So why weren't these used for very long after the war? The answer is that there were better options, such as air power, to delivery HE ordinance on fortified targets. Even by the end of WW2, it was clear that no reasonable amount of armour was going to protect an AFV going at a well defended position for a point-blank attack.
Now, all this said, there was one famous design that utilized this casemate design after the war, and that was the Swedish Stridsvagn 103. In this case, the idea was that since Sweden would be fighting an invasion (most likely from the USSR), it didn't need offensive capabilities. See, there was one thing I didn't mention that a casemate has as an advantage, and that's a low profile. These AFVs were designed to be able to lay low for an ambush on enemy tanks, and then quickly redeploy (with the ability to be driven at full speed in 'reverse' with a rear driver position). The turretless design also reduced costs, allowing a greater number of total units. Thus, this design represents a melding of pragmatism with an understanding of what the limits are. These units were retired once a Russian invasion was no longer likely, and replaced by AFVs more suitable to roles Sweden has in peacekeeping etc. The Russians and Americans did not need to make such compromises, and both of them, learning the lessons of the limits of casemate designs and recognizing the deminishing returns of total armour weight, went with more balanced designs, leading to today's MBT.
If you're interested in reading more about the development decisions behind these types of AFVs, Bryan Perrett's 'Tank Warfare' is a decent little treatise on the topic. Of course, there are a thousand books out there you can read about the choices the Germans and Soviets made in the race to upgun then uparmour each other.
https://en.wikipedia.org/wiki/Kanonenjagdpanzer
https://en.wikipedia.org/wiki/ASU-85
Both of these vehicles were manufactured in decent numbers.
If I may add: By 1943(and post-Stalingrad/Kursk), Germany was always struggling to produce enough war materiel that demand required, and this was only amplified after Allied bombing raids became commonplace after 1944-45. The Pz. III design was long outmoded and probably only useful to train new tankers, but the chassis was easy to build and maintain compared to the two Tigers or early Panthers. So, as others have said, tank factories just redesigned it to be a casemate tank destroyer - in some ways, making it even easier than before to build and maintain, as turrets are complicated things. But I'm no expert there.
Throughout the war, Germany had a policy of taking old designs(or, with the Jagdtiger, modern designs) and making them casemate TDs. This was because it suited the defensive nature of the war that Germany had; their infamous Blitzkrieg tactics through France and Russia earlier in the war were done with Pz. 35 and 38(t)s, and modern tank tactics - such as those witnessed in the Gulf Wars - are partly based on what the German generals did with those very early German tanks. In other words: tanks are best used as cavalry, which can go fast and strike hard, not sit around and wait in ambush. Germany's situation did not allow for that after 1943, so the casemate tank destroyers were an answer to soldiers desperately needing something armored with a big gun that can fight off attacks, not something to open breaches in the enemy's lines to exploit. That job was for the heavy tanks like the Tigers, when it was available. Which was rare.
TL;DR you want your tanks to be good on the offense, not defense, which is what the casemate TDs are.