Hmm...
I personally have never been stuck in a support role when I've chosen the feat (mainly because I only really use it when people are dying or when the situation is stable enough to allow for it). Also, I often find it difficult to not choose the feat or even to not choose it as my first feat (at first level even thanks to variant human). But then, I tend to play the guy you want to have the things you want others to have.
This is a "your mileage may vary" thing, I think. If you only use the feat as a last resort, then it would feel like a wasted feat to me. For our games, the healer feat gets used on all injured parties first thing after a combat so we aren't using potions or spells.
The games I've been playing in tend to feature a lot more combat and dungeon delving than socializing with NPC's, so taking a feat that doesn't help you fight or deal with traps or other dungeon puzzles feels like you are putting your character at a disadvantage. Most of the players want to contribute in combat, and not just by patching people up. Feats are a limited resource, so you want to use them effectively.
That being said, a group that has the healing feat is much better off than a group that doesn't, so usually someone will bite the bullet and take one for the team. Just as long as it's someone else, that's great.

(I realize that this is at least partially a perception issue, and one that you probably don't have.)
Instead, I'll note that I've always felt like it could be argued that the healer feat essentially allows the mundane materials and methods of a healer's kit draw from the energy of the multiverse to achieve its magic-like effects.
Yeah, the feat doesn't actually say anything like that. If that works for you, great. My group's consensus was that it was a non-magical skill, just abstracted the way hit points are abstracted.
Like a boss of bosses.
(Incidentally, one of my top two moments in tabletop roleplay involved a cleric of mine patching up a fully raging barbarian despite his (ultimately futile) attempts to kill her.)
Obviously, my group draws the "this is an acceptable break from reality" line in a different place than yours. We trade off GM'ing and one of the GM's felt that it was utterly ridiculous to apply medicine, bandages, and possibly stitches within six seconds while your patient is moving around and swinging a weapon or casting spells, and an enemy attempts to kill one or both of you. You are seem to be fine with that.

That GM convinced the rest of us to go along it (to try and keep some rules consistency between our games), and this house rule was the compromise.
1. Granting a feat for gaining a skill (especially one as good as the healer feat) seems like it would make the skill in question far more potent/useful than virtually any other.
Well, it still goes back to "only one person in the group needs to have this for everyone in the group to benefit from it, and doubling up on it gives no additional benefit". Now they can just choose to use a skill slot rather than a feat slot for it.
2. What if people want the Medicine skill but not the Healer feat?
Then they are being silly. But they can always choose not to use the healers feat.
Honestly, the first benefit is the more problematic of the two, if either benefit qualifies as problematic (neither does in my view). While 5-10 + Level/CR hit points a use is good, the ability to continually be able to get someone back into the fight is what I would have targeted.
Not at all. If they are back in the fight, the player gets to go back to being involved in the game, albeit at 1 HP so they should be at least a little cautious. This is the "acceptable break from reality" line I was talking about. Our group drew it right down the middle of the healer's feat.
For what you and yours are trying to do, it seems like a good change, though.
It's worked out pretty well for us so far, so I thought I'd share it here.