In my opinion, making sure the remastered game runs smoothly in modern systems should always be the priority, no one wants a 15+ years old game running with performance issues, but an updated control scheme is also very important. The ability to customize you controls can be a dealbreaker for a lot of players and surprisingly some remasters like the recent Castlevania Dominus collection don't offer that.
Besides that, I don't think they're needed, but I'm a big fan of those games that have a graphics toggle. It's a treat to be able to instantly compare how the game used to look with how it look nows. I played the Halo MCC games like that and it was very fun, especially because it has coop and we could promptly share our thoughts about the changes.
As for the examples of remasters done right, I'm following what others have already said, Nightdive is currently the golden standard of game remasters and their work with the Quake/Doom games should be studied by companies attempting to remaster old games.