I feel like stuff like the school scene was added because they had room to add some additional scenes during the process of streamlining the slower aspects of the games for adaptation purposes. I love the games but a straight adaptation of them wouldn't quite work since it would largely be Ryo running around talking to people without much else going on.
As for it being "better" than the games I don't think I'll view one as better than the other unless the anime gets really botched later on. For a good number of people though, the anime is just going to be easier to consume due to the divisive nature of the gameplay. I've never managed to convince anyone to try the games (and a few people I know won't go anywhere near them) but I already got two of my friends to check out the anime.
We'll see what happens with future alterations or changes in later episodes but for now I'm actually a bit excited to see what they do with it going forward. I'd be lying if I said I wasn't nervous about how the episodes covering Shenmue 2 will turn out, but it'll be interesting to see what they do with the material. My hope is that the anime will be good in it's own way. For now though the idea of this being a somewhat different take on Shenmue has me excited!
I do wish that they would have brought back more of the old English voice cast though, people can say what they want about the English voice acting of the games but it'll always be jarring hearing Ryo with a voice that isn't Corey Marshall's. Same goes for Lan Di as well, though Kyle McCarley's voice work for him in Shenmue 3 grew on me when I replayed it!
...Didn't intend for this post to turn into a long ramble but I always do this when talking about these games.