Thresh is a great jack-of-all-trades support. Any one thing he does, another supp does better.
Leona's CC is harder.
Blitz's pull is better.
An assassin/mage support's damage is better.
Braum's beefier.
He doesn't have sustain, like Sona or Taric.
I will agree that his lantern is pretty unique, but it's not going to save the adc every time; if he's far enough to pull his adc out, he hasn't been close enough to actually help in the fight the adc needs to be pulled out of. He can't engage a fight, and magically fly out; for that, see Kalista.
What IS good about Thresh is his ability to meaningfully contribute to nearly any situation. The CC of his kit's decent for singling someone out, or hampering everyone in a fight. His engage is good in a pinch, as long as he can hook/lantern someone to engage with him. His damage, especially with the passive on maxing flay, is mixed AP/AD, so it's always able to chunk some hp down in a fight. He's decently bulky, especially late with his passive, but he's no Sion.
Every support better than Thresh at something, and if your team needs that one thing more than anything else, that specialist is better. This is why he's rarely god or trash tier; regardless of whatever aspect the meta makes powerful, or makes weak, Thresh has some of each. Blitz, Braum, Leona or Sona might get nerfed or buffed, or shuffle around the tier list, but IMO Thresh is always just pretty good.
Personally I think he's one of the better designed champs, his kit has such neat synergy and versatility, without feeling gimmicky.