Let's compare Thresh with 3 other supports

First, let's compare Thresh with a healing support, Nami.

Nami Q can be used both to engage and peel, while thresh Q most of the times is only used as a engage. Problem is, Thresh hook is way easier to hit and more deadly, because he can use both his ult and E to make the target stay there because he can be on top of the person. While nami can use her ult and the slow of her E, her bubble is easier to dodge and her damage is lower.

Nami W is better for laning phase, if you compare to Thresh W. But ONLY in laning phase. Her W can be used to harass in lane or to heal her ADC, but the mana cost is HUGE and is only really good on 2v2 scenarios. Thresh's lantern can turn entire fights because he can save someone, or push someone to engage with him, while giving a shield and on top of that, is a huge reposition tool with a great range.

Nami E is great to add some damage and a nice slow for her ADC. But it only lasts for 3 AA and most of the times, it is better to use on a single target for maximum damage. Thresh E not only adds damage to his AA, to make him a bully support in lane, but also has a nice almost instant cast AoE slow, while also knocking up for a short time.

Only thing I think nami is superior to thresh is her ultimate.

Now, let's compare Thresh with a tank engage support, Leona

Her Q is basically a stun that resets her AA. Great for destroying wards. But she is heavily dependant on her E to make full use of her Q, while Thresh Q alone deals a lot more of things.

Her W is a nice armor/mr buff that applies passive in a area. In terms of dueling on a 2v2, her W is better, but again, only to this purpose. Anything else Thresh lantern is better

Her E is a engaging tool, that ignores minions, which is great to engage Healing supports. But not against Thresh because his E totally denies her E. And IMO, flay is better in most cases than her E.

Her R is better as a engaging tool than his R, but his R is more of a peeling ability or a ability to hold on for more time his targets. Her R stunning range is short, so to have a maximum use of it is kind of hard.

Now let's compare Thresh with a tank peeling support, Braum

Braum Q is nice to apply a stun , assuming you proc his passive, and also to harass in lane. But Thresh is able to harass with empowered AA from his E, plus his Q can not only stun, but make him able to jump on someone if it connects.

Braum W is worse than Thresh one, IMO, in pretty much any case except on a 2v2 fight.

Braum E is nearly impossible to compare with thresh E. They are good for peeling in two very different ways, but thresh E is also good for engaging.

Braum R is better than Thresh nearly always

Tl;dr Thresh may not be the best at doing one thing or another, but he does everything way too well. He needs a utility nerf