In what sense are Persuasion and Intimidation negative? Wouldn't that depend on what they're being used for?
They might be 'for your own good,' in theory, but they don't actually offer any sort of mechanical benefit or 'carrot' for going along with them (a DM might offer inspiration of it was in-character for a certain use of Persuasion or Intimidation to affect your PC for some reason, I suppose).
To be clear, when you say they don't offer "any sort of benefit," you mean any sort of mechanical benefit, right?
We are talking about a mechanic - a persuasion or intimidate check - yes. You might try to persuade or intimidate a character into taking a course of action that's "for his own good" but against his nature, for instance. You're essentially offering an IC excuse for the player to take what he probably knows is the better course of action, an 'out' from RPing to the point of stupid, I suppose. You could consider that a non-mechanical 'benefit' - or not, depending on how you felt about it.
Again, you mean a mechanical positive, right?
Certainly mechanical and probably narrative. Intimidation is pretty clearly negative in concept as you're threatening, persuasion can be (you can appeal to fear, greed, etc as well as more positive emotions or ends), while inspiration is a positive thing by it's very nature, at least for the one being inspired (though, you can re-skin or RP things how you like: an Orc Warlord might 'inspire' his troops by threatening to torture them).
Do we all agree that there is social pressure on players not to use spells or skills that influence PCs' minds or feelings without the consent of the targeted PC's player?
It's an area that can be touchy for some players, but it's far from the only one. There are many potential sources of friction among players, and some classes, like the Paladin (or any 'religious' class) or Thief or Warlock, or any caster, really, can be at the center of it. It's not a great reason to exclude classes, and if you did, you wouldn't have many left. Fighter(Champion), prettymuch'd be it.
Is it also fair to say that there is social pressure on a player to accept a proffered mechanical benefit to a PC?
A little less fair than it is to say that there is social pressure on a player to proffer needed mechanical benefits to the other PCs, but, yes, certainly. It's a cooperative game. The pressure on a Cleric to heal allies (even if they're ungrateful heathens) is greater than the pressure on the allies to accept being healed (even if they despise the Cleric or his deity for some reason). Doesn't mean there shouldn't be Clerics or support classes in general. Parties need support contributions to succeed, so there need to be classes that can make adequate contributions in those areas. Ideally, they should also do other things. A War Cleric or Valor Bard can handle some melee, as the Warlord could, for instance, and with the extreme flexibility of neo-Vancian casting, all the existing support classes can easily make alternate contributions very effectively, as well. The 5e martial options need to be able to move beyond the current tanky-DPR and DPR/exploration-skill-monkey contributions, and the Warlord is a prime candidate to get that ball rolling with a non-caster alternative to the existing support classes. Heck, with 5e's expanse of design space, it should have some modicum of their flexibility, as well - something that'd be very appropriate to a class that includes tactics in its bailiwick, as tactics have to be adaptable...