How do you define "best results" here though?
It sounds like you think that there is a "correct" way for a player to have his or her character respond to something. Would you clarify?
In addition, we have a mechanic for encouraging players to act consistent with their personality traits, ideals, bonds, and flaws - Inspiration. I like that it's a mechanic to reward players acting a particular way if they so choose rather than one that is used by the DM to tell them how to act (e.g. "intimidated").
I have played with many groups and many players, and most, myself included, have one thing in common: they think they know a lot more about the world, their characters, and the campaign in general than they really do, and most of the assumptions players make are generally at least partially wrong. This has really only lessened when I've managed to play with the same people routinely for long periods of time, and even then, it can be surprising what assumptions others are bringing to the table. Usually the moments when those assumptions are most likely to come into play is precisely the moments that you are advocating leaving everything up to the player. This causes a lot of stress at a lot of tables. This is where using some of those mechanics on the PCs can help, if used right. You don't tell them them, "You're intimidated, take a -2 to all rolls for the rest of the encounter," you tell them "you are intimidated because (flesh out world or character history here to give them something to work off of), take a -2 to all rolls for the rest of the encounter."
Inspiration is a great idea, but generally requires the player to actually understand enough about the world, character background (i.e. what does being a noble of the city of Israth really mean?), how they fit in the world (do they agree with the general behavior of a noble of Israth or do they deliberately do something different?), etc. before that mechanic can really have full effect. Short of sitting everyone down and giving a long explanation or writing it all out, and hoping they take the time to read it, the moments where any of the skills can potentially be used are the only real chances a DM has to actually get across enough of this information for player choice and inspiration to really work. It sounds like your group can do this without a lot of heavy reliance on mechanics; most groups aren't that lucky. Most groups I've been in, that really doesn't happen; some or all of the players would play off the intimidation as a failed attempt, the scene largely ends up being a largely blah encounter, and the game suffers. There really does have to be some kind of mechanical oomph behind the perceived negative as most players will not touch anything that leaves them less than 100% total control, and therefore, never really grow either themselves as a player or their characters. It's the same reason that most negative stats tend to be downplayed by the majority of players. Any kind of exposure or weakness in the character is actively and quickly tamped down, leaving little room for the DM to expand the story, world, or characters beyond what the players currently know (or think they know) without a lot of force. Much of this is well earned on the player's part, as many have had bad experiences with DMs that went overboard, but ignoring the mechanics completely isn't going to resolve any bad experiences.
In the end, for those lucky enough to find a consistent group to play with, simply ignoring or choosing to not apply rules is a perfectly valid option because it can be done in a larger framework of trust and common understanding of what the game should do. For most, it isn't; sporadic and selective use of the rules when trust and common understanding is not already present, as it won't be in most pick up groups or organized play setting, actually makes things harder, not easier. Trust and common ground only come with consistency, and folks willing to bend the skills rules so far as to not apply them to PCs are generally also going to make other changes to the game that newcomers are not likely to fully understand and immediately appreciate. There's a reason I won't play 5E with just anyone, or for that matter, any organized play league now that I have other options; too much potential table variance makes it hard for me to understand precisely what I can reasonably expect in most cases. So, while your approach probably works fine for your table, it would absolutely collapse at many others. This is both the strength and weakness of 5E's making the DM the center of the game.