I think the gladius was used like a pestle or meat tenderiser for stabbing downwards, or diagonally downwards, BEHIND THE SHIELD on those that the Roman shield, and behind it a sort of scrum formation, had knocked to the ground.

They may have used both a backhand and forehand motion. If they were leaning into their shield, then as well as providing more power and protecting their feet, their thrusts would be downwards and diagonally forward to allow a diaphragm piercing abdomen to ribcage thrust.

The Roman style evolved out of combat with the Greeks and Macedonians who used tight phalanx formations with long piercing lances. The most common Roman lances, pilum, however where single use throwing weapons.

The Macedonians beat the Greeks by using longer lances. The Romans beat both by creating a phalanx with no forward facing weapon. They beat both, and subsequent barbarians, by concentrating on the crush. They probably realised, from looking at phalanx warfare that it was not the lances that did the most damage, but the crush, being stamped upon, and impaled subsequent to one or other side given way. The foot stomp was augmented 100 fold by the gladius as stabbing pestle.

The clue to this use comes in Seleunius's short speech to his legions before the battle of Watling Street, where about 5000 Romans massacred about 80,000 Britons with a loss of only 400 Romans. If the Romans had fought manoamano in any fashion, even in formation, they would have lost far more than 400. The agility and strength of their fresh opponents would have resulted in greater loss.

Seleunius said "knock them down with your shields and stab them with your swords." That is the inhuman way that Romans fought. Realising that in Greek/Macedonian style battles more of the carnage came, not from being lanced, but the crush of the crowd, the Romans concentrated on killing by shield, crush(bowl over), and stab.

Shields on their own do not kill all that well but
1) if you can first anger your enemy, by throwing pilums at them, getting them in a vendetta kind of frenzy, and
2) with only a short sword appear to be an easy target,
3) you can get a throng of nonRomans to crowd up close and vertical (thus unable to push hard,
4) then you can close your shield wall, get down low and diagonal to your shield,
5) and with drum and trumpet sounded organised push their shield wall forward,
6) resulting in the collapse of opponents who appear under the shield wall feet first.
7) If you then hold your gladius backhanded in front of your chest like you are using a pestle, or at your side forehanded, and dig down at the feet, thighs, crotch, abdomen, and finally diaphragm of these collapsed foes, as they appear beneath the shield, they will be bleed out and be dead or asphyxiated before their hands or faces appear before they can attack.
The Roman phalanx was a giant croupier's rake combined with a meat tenderiser: a Giant Roman Roomba.

If you have been in a crowd, such as at a football stadium, then you'll know how scary and dangerous they can be.

The shortness of the sword was threefold
1) useful as pestle for stabbing those they had pushed over on the ground with their shield,
2) making legionaries look less dangerous so opponents would approach so they could be pushed over,
3) also forced the legionaries to cooperate with each other and remain in formation. The fact that it was pretty useless in the ways described by the Smithsonian video, forced the legionaries to use their cooperative, shield, scrum, formation techniques.
There should be evidence of this hypothesis in history but I can find no mention of pestle. Some evidence.
1) The handle of gladius often has a pommel to enable putting ones weight on it to push down harder in cases where the failed foe had some armour.
2) The scutum page on Wikipedia shows a carved relief of a legion at work trampling on a living foe, which suggests that the killing went on behind the Roman shield not in front of it.
3) The fact that those further back in the formation were equipped with lances suggests that the stabbing went on behind the Roman shield wall not before it.
4) The thickness of the blade is wider than other stabbing swords. This may be because it was for cutting as well (making it neither good at stabbing nor at cutting) but I claim the thickness was not meant for quick stabs to a moving enemy, but to downwards and diagonally downwards pestlelikemovement , and sometimes downwards stabs using the weight of the legionary on those crushed and prone on the ground. It needed to be thick so as not to break.
5) The way that the Romans committed suicide, by falling on their sword piercing their diaphragm and heart, was probably the way that the used their swords in combat. They used their weight to push the sword into the rib cage of prone opponents, and the same method when they killed themselves.
6) The Italian gesture for "up yours" mimics the use of the gladius.