At the crux of why there is more connection to prewar than postwar is just how much the show needs to introduce to a new audience. It's not easy to build up a setting in such a short amount of time without compromising the characters' stories. The audience might know 1950s America, the cold war, and nuclear armageddon as familiar shorthand, but they do not know a constitutional Republic with a 2 headed bear on its flag or how advanced robots powered by nuclear technology work together with it. That shit takes time to set up, and that lore can never be the backbone of a show; any show.
Lore cannot be dumped upon the audience because it isn't what people engage with stories for. The lore needs actual characters who you actually care about to get you invested in lore in the first place. Stories that don't put characters first not only don't sell as well, but also aren't very good. Games can cheat a bit by having you be the narrative thrust and everyone else be minor characters, but any non interactive medium does not have that luxury.
What I mean:
The story of the NCR or the Legion or House is not built on how interesting their factions were in the game, but how interesting the stories you experience because of them were. A young woman you save who then goes on to build a nation. A charismatic Howard Hughes type and the people trying to make it out on top in his orbit. A psycho wearing an animal head and the man who conquered dozens of tribes into a brutal empire despite not being the smartest or most traditionally charismatic. Hell, the reason the Brotherhood is so much less interesting is how often it doesn't have as compelling characters.
All this is to say that it's a miracle the lore comes out as well as it does. Most of the places where it was compromised was so it wouldn't take up more time in the relatively short 8 episode runtime. Directly showing Shady Sands destroyed conveys the fall of the NCR far more quickly and more effectively than anything else. Having a deeper reference to One For My Baby than Cooper misremembering Carla's name and how long ago she was there wouldn't have been worth it for people who didn't play New Vegas. Them not making these changes would've hurt the story they were telling, and that is never a good idea.
Another problem that isn't actually a big problem
is people wondering how the former NCR areas are in such bad shape. I think people missed just how empty and desolate most of the NCR was, even at the best state they were directly shown. 700,000 people in the Area of California is not that densely populated, as California right now is huge and surprisingly rural in many areas. The distances you travel in Fallout 2, as someone who grew up in California, are no joke. Not seeing as many signs of it being rebuilt in the show makes sense if most people lived in smaller population centers for safety and many of them fell along with Shady Sands.
A similar problem comes from New Vegas, with the state it was shown in game being deceptive. House built up the facade of the Strip like a decade before the start of the game, with the perception of rebuilt civilization existing so House to demonstrate power he doesn't have. New Vegas also led fans to believe things were more rebuilt than they were, but even in lore it was deceptive. Factoring in the way space gets warped in the 3D games, the west coast was far closer to post apocalypse than post-post apocalypse.