Games

The show is good for personal drama, but terrible for anything about logistics and they really have a hard on for the brotherhood.
The NCR was somehow wiped out because a single city (which was depicted like it was some minor suburban farmstead rather than a sprawling metropolis) got hit with a Fatman nuke.

Similarly Caesars legion is depicted as two camps of maybe a 100 guys each, when that faction was fucking massive, containing most of the Midwest. Consisting of 86 different subjugated tribes - which should, at least for a good bunch, moreso be thought of as small nations.

The show, likely because Bethesda is like this, depicts a post-apocalyptic hellscape, despite Armageddon occurring several hundred years ago. Fallout isn't post-apocalyptic. It's post-post-apocalyptic.
The two first games and new Vegas show a world moving on, nations developing, there's talk of trade and roads and standing modern armies. Of course the scale is kept at a level computers could handle, but it was always indicative of something much larger. Mr. Houses securitron facility gives a sense of the real scale. As does the concept art for Freeside.

Caesars legion was doomed to fail, but not because it was teeny tiny.

It was always silly how they were somehow a threat when they were just larpers in football gear, but it was also made clear in the game that they were only really a threat because of corruption in NCR, and most of Caesars plan for victory depended on a lot of guerilla tactics behind enemy lines.
Part of that larp is also meant as critique. Caesar is a fucking loser idiot, mildly more competent than his competitors were. Like any mediocre man he gets delusions of grandeur and suddenly we are reviving Rome (which is somehow a refutation of the old world? Caesar is a dumbass, it's so well written, I love it.)

edit: You might find some answers to your questions in this thread

The show makes the Legion out to be a bunch of dweebs larping in the desert.

That's because they are. Caesar was an educated NCR citizen (an anthropologist or something, I don't remember), who through cunning and unseen brutality, and knowing to surround himself by much more competent people (his legati), managed to unite a ragtag group of warring "tribes" in Arizona. He larped Roman stuff because he was a nerd who found some old books about Rome in a ruin, but at the end he was just a more successful warlord.

By murdering or enslaving everyone who opposed them, and by employing basic war strategy and logistics they overwhelmed every other group East of the Colorado, covering a lot of land really fast, fast enough to concern other regional powers, like Mr. House or the (not yet nuked) NCR.

If the courier (the protagonist of F:NV) is smart enough, they can realize that this blitzkrieg-like strategy is unsustainable in the long term, and if the legion keeps conquering territory it will collapse under its own contradictions. The only thing keeping middle ranking officers from rebelling is Caesar's leadership, and that the rate of conquest outpaced the stresses of governance. Caesar has also killed or kept down some of his smarter generals, so he's left alone at the top of the legion. Should the legion stop expanding, or Caesar die, it would fragment immediately into factions vying for power.

In the show, Caesar has died years ago, and two factions claim to be his true successors, but it looks like they don't really control territory the way they did when Caesar was alive, because they've gone back to warring amongst each other, and because these paleo-fascists can't keep state power, since all they know is conquest and brutality.

They've always been larping dweebs, who happened to be a little better organized and had better leadership than their opponents, until they weren't anymore. The way they're portrayed in the show just shows what all the pageantry is without any actual state power.

Part of the story of New Vegas is that the Legion is stretching themselves very thin trying to take over Hoover dam and leaving themselves vulnerable, if they lose the battle of Hoover Dam in the canon it's pretty much implied they will be almost entirely destroyed.

So is the show depicting them as much less impressive than they are in the game

I think the show writers just don't understand scale, which is why the legion which was supposed to control like half the midwest past the hoover dam is just like 36 guys pointing guns at each other, and the Brotherhood of Steel which is supposed to be, yknow, kinda weak has like a dozen chapters of hundreds of dudes each with a fucking flying aircraft carrier

Their main territory was located east of Las Vegas in Arizona and New Mexico, the so called 87 Tribes; like the NCR they were overextended in Nevada, in fact most of the factions in New Vegas are overextended in some manner which is why the player character has such a outsized impact on the politics of the region