The definition of 'important' is kind of subjective though, which is the issue. Some people loved Pokemon following you around in HGSS, difficulty settings in B/W--hell, the fact that HGSS had its menu always set on the bottom screen was rad as hell. But they all got rid of that.
They don't cut things that directly affect gameplay, but I think it's more than fair to say that they cut enjoyable features if they don't deem them important. They don't get rid of anything that affects the main gameplay, but everything else seems to be fair game to them.
One of the coolest feature that Game Freak added in the Johto remake was the option to auto run without having to hold B which made training so much easier but for some reason they just dropped it. Though I wouldn't mind if they dropped the roller stakes, those were a horrible idea.
Personally what I want them to do is to go back and update some of the texts such as the dialogue when teaching a Pokémon TM due it being the same text from the original gen 1 game.
There is no need for the dialogue to tell the player what the TM is as Game Freak fixed that in later generation.
For example the old text would say;
"Booted up TM 65 Shadow Claw, teach a Pokémon Shadow Claw?"
Then if a Pokémon already has 4 moves.
"Charizard already knows 4 moves, delete a move for Shadow Claw?"
"1 2 3... Poof Charizard forgot Ember and learnt Shadow Claw"
I would have it that it says;
"Teach a Pokémon Shadow Claw?"
"Delete a move for Shadow Claw?"
"Poof Charizard learnt Shadow Claw"
The second option would make swapping TM less of a chore as you wouldn't be subjected to long lines of texts.
but the menu thing is back in Gen VI, in fact much better even
Making them tiny icon isn't what I call much better.
I'd argue pokemon-amie does following in a much better away
While Pokémon Amié does a great job in allowing the player to bond with the Pokémon when they are resting but it is no substitute for Pokémon following you as the Pokémon in Amié don't react to the environment in the over world.