Well, I'll start by admitting that Thresh is simply overtuned, his kit is overloaded, and he could use a real nerf. Most preferably, put some sort of delay or some ability to reliably interrupt his darn lantern.
I don't care about win rates or what anyone says on the matter really. His lantern is just OP as it allows way too much safety. I can deal with his hook, flay, and ultimate. His lantern (like Tahm's consume) is just stupidly annoying though.
To the point though:
I actually hate having a thresh support (as an ADC main). I don't mean to sound arrogant, but I typically win all of my early trades as an ADC. I typically dramatically outfarm my opponents as well in the early game. I use this to build an early lead even without kills and use that to snowball the game.
So what happens when I have a thresh support? His engage is unreliable. Sure, if he gets lucky and lands a bunch of hooks I love it, but in general every time he misses a hook that's like 20 seconds where pressure is against me bot lane and the enemy support gets to harass me for free. I don't like it. I'd much rather have a Leona or Braum who can engage more reliably and still be relevant if they miss the skillshot. And I say gets lucky because, outside of the unusual scenario where the enemy gets in a really bad position relative to the wall and minions, landing a hook is as much about the Thresh player being skilled as it is about his opponent being unskilled at dodging it.
The other thing that happens is Relic shield. Thresh needs to actually last hit minions to get the procs. With a thresh support you need the healing on the proc because he offers no other sustain and as I mentioned has an unreliable all-in. Most thresh's seem to only have about a 60% CS rate trying to execute the relic shield unless you set up a minion for them that the wave isn't attacking. So I do this (which costs me time that I could be harassing) or I let him try to time his auto on the cannon and make me miss 40% of all cannon minions and a lot of melees to boot. I don't like it. When I'm regularly hitting 90% CS in lane I don't want a support trying to last hit, it really throws me off a LOT. Again, a Braum or Leona is going to get that last hit long before there's minion threat and I miss nothing. Much preferred.
I also prefer to play against Thresh over Braum / Leona or even Soraka / Sona / Janna / Taric (the former 2 because I win most trades and their sustain sort of kills that, Janna is safer during laning phase, and Taric has a heal and a targeted stun). That said, I prefer going up against pretty much any of these champions than Thresh or Tahm Kench in the late game as both of these champions offer their allies way too much safety.
One of the worst feelings in League is finally catching the fed enemy out of position and almost killing them, only for a Thresh lantern (or TK consume) to get them out to safety, when you KNOW they weren't counting on that (i.e. the Thresh was far away when you got the pick and simply made it there before you managed the kill).