It is fair to say that Roman infantry combat was more sword centric than Classical hoplite warfare or Hellenistic Macedonian pike phalanx tactics. The case for the sword as a national weapon has been argued by Simon James Rome and the Sword (2011). But one thing that is important to note is that Romans did not just fight with gladii. Rather, the sword worked as part of a "weapon system" that consisted of the soldier's pila, scutum, and heavy body armor. Skallagrim's video makes good points, but note his swordsman confronts a pole arm bearer when armed only with a sword, and is also alone. Roman legionaries were effective with their swords because they were carrying a large body shield, wearing a thick cast bronze helmet and by c. 200 BC, mail shirts; and when they attacked enemies with their swords, they may have just softened them up with heavy shield-piercing javelins. In those conditions, a good sword like the gladius hispaniensis works pretty well.

The first key complimentary weapon was the pilum. This was a heavy javelin, whose long shank allowed it to punch through a shield but continue to penetrate forward to reach the man behind it. Javelins were very common in ancient warfare, but usually as a tool of light infantry (and by light infantry I mean infantry that was not expected to hold ground, but fell back to avoid close contact with the enemy.) Most javelins were however too light to pierce a well made shield, and were therefore primarily skirmishing weapons. But the pilum allowed Roman hastati and principes to engage enemy heavy infantry with javelins, obviously a real advantage. Who cares if an opposing phalangite has a 20 foot sarisa, when you can chuck a pilum at him from 75 feet, and pierce through his shield. So trading a spear for a heavy javelin offered some real tactical advantages.

There is one downside of arming heavy infantry with javelins: they don't have very many! Roman legionaries carried two. Not a lot of ammo. So if they are expected to hold ground, they need to be able to fight with their swords for extended periods. The gladius, rather than being a backup weapon of last resort, was therefore a primary weapon.

Next the gladius worked well with the scutum. As the query notes, swords are short. Roman gladii during the Republic on the whole seem to have been about 10-15 cm (4-6 inches) longer than Greek xiphe or machairai (although they shortened under the empire), but even so a 65 cm blade still doesn't have all that much reach. So you need a good shield if you have any hopes of getting close enough to a spearman/pike man to get inside his shaft and do damage with your blade. And the scutum is a very good large body shield (c. 4 feet by 2.5 feet). Furthermore, because it is a boss-gripped, its possible to punch forward with it. If you use the lower rim, you punch about 4 feet in front of you, giving you extra reach beyond the two foot blade. Knock your enemy off kilter, and then close to finish him with the sword. The tactic of punching with the lower rim can be seen on the Pydna monument of Aemilius Paullus, on an early imperial metope from the Mainz Principia, and is attested literarily by Tacitus (Agr. 36).

Finally, Roman legionaries were very heavily armored, and this gave them an additional advantage is close in sword fighting, where again you need to survive getting inside your opponents' spear and its best if every blow they do land is not lethal. By the late 3rd century BC, Roman soldiers were the first Mediterranean people to widely adopt mail armor, and this likely offered a special advantage when fighting more lightly armored opponents (B. Devereaux, "The Adoption and Impact of Mail Armor in the Third and Second Centuries BCE" Chiron, Forthcoming). Even before that, the Montefortino helmet was made of very thick cast bronze, putting a lot of metal between an enemy's spear and the Roman's skull. So when you are very heavily armored, you have both the physical protection and psychological confidence to close with a two foot long gladius. And your more lightly armored opponent, who has a shorter sword and smaller shield, is going to be all the more spooked the closer you get.

The closest thing the Romans encountered to a bladed pole arm (i.e. excluding spears/pikes) was the Dacian falx, a long sickle attached to a pole and wielded with two hands, sort of like a late medieval bill. The solution in the Dacian Wars was simply to increase the armor of the legionary, with reinforced helmets and the addition of gladiatorial style manciae (sleeve protectors), still making them superior in close combat to Dacian warriors.

Finally, Roman soldiers did not fight alone. The length of the Roman gladius, still relatively short, was largely determined by the need to maintain relatively compact infantry formations, although the Romans famously fought in a looser order than the Macedonians. But still, the legionary was a soldier in formation, which means if he got into trouble, there were soldiers next to him and behind him who can bring their gladii/pila/scuta to bear.

The Romans did not necessarily prefer swords over polearms, but used it alongside polearms in different ways depending on the timeperiod. The Romans used polearms as a primary weapon and as a part of a primary weapons system (a combination of sword + thrusting spear or sword + throwing spears) throughout their entire history (including both the classical Romans from the 700s BC to the 400s AD and the Eastern Roman Empire which continued to the 1400s AD). The sword was never a primary weapon all by itself - it was always used in some sort of weapons system along with other primary weapons.

The earliest Roman armies were classical phalanx hoplites equipped with long thrusting spears. After manipular reforms that divided troops into maniples (and groups such as velites, hastati, principes, and triarii), spears were still retained as heavy thrusting spears and as multipurpose throwing spears. Polybius describes hastati as equipped with two pilums (which were long throwing spears that were ~7 feet long and could double as a thrusting spear). Polybius also describes hastae (long thrusting spears) being used by triarii as their primary weapons: "The common soldiers wear in addition a breastplate of brass a span square, which they place in front of the heart and call the heart-protector (pectorale), this completing their accoutrements; but those who are rated above ten thousand drachmas wear instead of this a coat of chain-mail (lorica). The principes and triarii are armed in the same manner except that instead of the pila the triarii carry long spears (hastae)." -The Histories (Book 6) by Polybius Http://penelope.uchicago.edu/Thayer/E/Roman/Texts/Polybius/6*.html

The Principes before the time of Polybius were also equipped with thrusting spears before they were reequipped with the pila. The Hastati were also occasionally reequipped with the thrusting spear in certain occasions, see quote: "The Tribunes accordingly gave out the spears of the Triarii, who are the last of the three ranks, to the first ranks, or Hastati: and ordering the men to use their swords only, after their spears were done with, they charged the Celts full in front. When the Celts had rendered their swords useless by the first blows delivered on the spears, the Romans close with them, and rendered them quite helpless, by preventing them from raising their hands to strike with their swords, which is their peculiar and only stroke, because their blade has no point." http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0234%3Abook%3D2%3Achapter%3D33

During the late Republic, the reforms around the time of Marius that equipped most legionaires with pilum and gladius simply resulted in the core heavy infantry troops switching from a more dedicated thrusting spear to a more multi-purpose throwing spear that could also be used to thrust (pila). The gladius was used in conjunction with this throwing spear (that can double as a thrusting spear) as a part of an overall weapon system. See below in regards to the myths of the bending pila and the pila used in many occasions as a thrusting weapon. Swords/gladius were never used alone by themselves. So even during what we consider the heyday of Roman sword usage, the sword was still a part of a weapons system where there were multiple primary weapons.

By the late Roman Empire, the pila was supplemented or replaced by war darts, and heavy thrusting spears made a come-back as the Romans fought more and more cavalry opponents (a dedicated thrusting spear was still likely better than a heavy javelin/throwing spear that could double as a thrusting spear).

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Pila as a thrusting spear and the myth of the fragile pila that could only be used as a ranged weapon:

There is a lot of evidence that the Romans used the pila in melee combat as thrusting spears (so they were sturdy): Plutarch describes Caesar's men at Pharsalus jabbing upwards at the faces of Pompey's cavalry with their javelins. "And this was what actually came to pass; for they could not endure the upward thrust of the javelins..." -The Life of Julius Caesar by Plutarch

"Whenever the cavalry charged, they were to run out through the front ranks, and were not to hurl their pila, as the best soldiers usually did in their eagerness to draw their swords, but to strike upwards with them and wound the aces and eyes of the enemy" -The Life of Pompey by Plutarch

"But Caesar, observing that the left wing of the enemy was enclosed by such a large body of horsemen, and alarmed at their brilliant array, sent for six cohorts from his reserves and stationed them behind the tenth legion, 3 with orders to kept quiet and out of the enemy's sight; but whenever the cavalry charged, they were to run out through the front ranks, and were not to hurl their javelins, as the best soldiers usually did in their eagerness to draw their swords, but to strike upwards with them and wound the faces and eyes of the enemy; for these blooming and handsome war-dancers (he said) would not stand their ground for fear of having their youthful beauty marred, nor would they face the steel when it was right at their eyes." - The Life of Pompey

Antony's invasion of Parthia: Plutarch in Life of Antony 45 talks about Mark Antony's legions using their pila to thrust at Parthians in melee. "But the Romans, with a full battle cry, suddenly sprang up, and thrusting with their javelins slew the foremost of the Parthians and put all the rest to rout." -Life of Antony by Plutarch

In this incidence, pilas would definitely be of primary importance and much more useful in melee than their short swords, We have reliefs on Roman artwork showing Romans using pila in melee combat stabbing at Dacian infantry. See Tropaeum Traiani

Caesar's writings about Alesia are translated in some sources as saying he had his troops first use pilums as spears/pikes before switching to swords. "omissis pilis gladiis rem gerunt" -Gallic Wars Book VII

Arrian in Array against the Alans (2nd century AD) talks about legionary heavy infantry equipped with an iron shank weapon used to thrust at horses: "Fifteenth legion's infantry should hold the entire right center above the middle of the whole area, because they are by far the most numerous. They should deploy in eight ranks and their deployment should be close ordered...And the front four ranks of the formation must be of spearmen, whose spearpoints end in thin iron shanks. And the foremost of them should hold them at the ready, in order that when the enemies near them, they can thrust the ironpoints of the spears at the breast of the horses in particular. Those standing in second, third an fourth rank of the formation must hold their spears ready for thrusting if possible, wounding the horses and killing the horsemen and put the rider out of action with the spear stuck in their heavy body armour and the iron point bent because of the softness. The follow ranks should be of the javelineers." -Array against Alans by Arrian

Finally, in regards to the durability of the pila, it is debatable whether the pila was actually intended to bend, or whether it bent due to other situations (such as improper removal of a pila stuck in an object). More modern evidence suggests that the pila could get stuck in a shield without bending, and it was not designed to bend. We do know that the Romans used lighter javelins that were more likely to bend and also used heavier javelins (eg. different types of pila) that were less likely to bend. For example, Polybius states that the light, 3-foot javelins used by velite skirmishers had thin points (hammered thin) that were more likely to bend on impact. However, he contrasts this against the 6-7 foot pila, and states that the pila had a stronger design and how well the iron shank and points were firmly attached the the shaft:"Each is fitted with a barbed iron head of the same length as the haft. This they attach so securely to the haft, carrying the attachment halfway up the latter and fixing it with numerous rivets, that in action the iron will break sooner than become detached, although its thickness at the bottom where it comes in contact with the wood is a finger's breadth and a half; such great care do they take about attaching it firmly."

Scholars such as MC Bishop originally thought that the pila was supposed to bend on impact in his older works. However, in his more recent work, he revised his opinions in light of new evidence to say that the pila was probably not intended to bend on impact. According to his more recent works, such his “Pilum: The Roman Heavy Javelin,” many if not most Roman javelins are actually very sturdy and don’t bend on impact (with exceptions such as Marius’ wooden peg pila or the Velite’s very thin javelins). He states that most pila that do get bent are bent from people improperly remove a pila stuck in another object by applying to much lateral force while trying to wiggle it out. The pila is perfectly capable of being stuck in a shield because of the shape of its head without needing to bend. There are plenty of experimental archeology videos on Youtube where people test the strength of the pila in close combat, and found that it works as a thrusting spear quite well.