How much does system matter for the players to enjoy themselves?
My opinion is that the most important thing in order to have fun is the playstyle rather than the rules strictly, but it's quite clear that the rules can support one playstyle well well and get in the way of another playstyle instead.
A playstyle is not so easy to define, because it's made up by a myriad of elements, e.g.: your expectations about character death, the level of realism provided, not breaking suspension of disbelief, the amount of math calculations required to resolve an action, complexity (e.g. the amount of stuff about your character you have to keep in mind at all time), open-endedness (i.e. the balance between what you want your character to do and what the rules will tell you to do instead) and many more.
I also agree with those who say that a rules system tends to matter more in terms of going against your playstyle rather than supporting it.
The DM has limited power to make a system work better, for at least two reasons: (1) normally a system is designed as a whole, every house rule has a chance of causing unwanted consequence requiring even more house rules; (2) every player effectively has different preferences, so making the game fit all of them can be impossible.
Players should responsibly accept that not every one of their preferences can always be granted, and it's best for a player to look at the bigger picture and try not to be bothered by minor "flaws" of the system. It helps a lot to remember yourself, that if you can't have all your pet peeves satisfied in this campaign, there is always the possibility of playing another campaign later.
Then of course, it's a matter of gathering good info about that "bigger picture" of each rules systems. What was the purpose behind system XYZ? Is the system widely considered to succeed at that purpose?