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