I think Yellow did a pretty good job of following the anime initially, but really fell off as the game progressed. Pikachu following you was a nice touch (though I'm still not entirely sure why the rival started with Eevee, rather than one of the standard starters like in the TV show), and they did pretty well with giving you the other starters (Bulbasaur from a girl in a house, sort of like the Hidden Village episode, Charmander from a guy at the top of Route 24, and Squirtle from a police officer). They could have incorporated longer plotlines for getting the starters, I guess, to make it more like the anime, but I think what they did there worked well enough.
Having Jessie and James show up at random points worked well at first, but it didn't mirror the anime as well as I would have liked. I would have preferred to encounter them more frequently, and I would have liked to see more of their personalities like in the TV show. Also, they seemed to disappear partway through the game, never to be seen or heard from again. It would have been nice to have them play a more integral role in the game, especially in the late game. They also could have had Meowth follow them (kind of like Pikachu follows your character).
I don't think it would have been practical to have Brock and Misty follow you throughout the game, but it would have been nice to have them have more unique dialogue in Yellow. Perhaps they could have been trainers you returned to for advice later in the game, or maybe they could have been tied into some sort of side quest.
As for the Elite Four, I guess they could have used a format like the PWT in gen V. You could still have battled the same trainers each time (they could have put in trainers to match the ones Ash faced in each round, with Richie as the champion, so to speak). But the issue is, at the end of the day, Ash never wins a league tournament, and that conflicts with what most players want to see in a video game. Most players ultimately want to win, and conquer the Elite Four/whatever then "final boss" is, and doing that would have meant diverging from the path the anime followed. So perhaps a tournament would have been a nice idea, but it could never have realistically stayed true to the anime in terms of having the same outcome as the TV show.
Ultimately, Yellow feels more like R/B with an anime flavor, rather than a true imitation of the anime. As an anime-tinted version of the original games, it does pretty well, as there are a lot of anime elements that would have been difficult to incorporate into the game while keeping the same basic mechanics and structure. A fully anime-based game would have been cool, but that would have taken a lot more work on the part of GameFreak, and I get the sense they mainly wanted to capitalize on the momentum of the Pokemon craze in the most time-efficient and cost-effective way possible.