actually, his q doesn't displace the enemy that much. it's more intended as a gapcloser/cc ability than a displacement.
his e isn't a stun, it's a disruption. big difference there (and it's actually surprisingly easy to screw up. I didn't believe it until I started playing him).
don't forget, neither of his abilities are teleports, theyre gapclosers. big difference there as well. one is instant and lacks any defense while the other can be interrupted and has reaction time. (also, you can body block the lantern if they use it defensively).
his "mega aoe slow" is no stronger than a zilean w. it is technically aoe, but in reality you're unlikely to hit more than 1 or 2 people with it because the gates go down after being hit.
melee supports don't have poke because they are meant to go hard or go home. almost all of their kits revolve completely around that. besides, when thresh goes out to poke he leaves himself VERY open to counterattacks.
saying that he's a ranged champ so armor/mr doesn't factor is like saying that a whale being able to hold its breath for a long time means it can stay under indefinitely. safer doesn't mean invincible.
not sure where you're getting that thresh has tons of mobility from. he has no gapcloser aside from 1 initiation ability (and his w if youre counting for allies). it isn't super low (necessarily), but it certainly isn't a lot.
now lets look at the champ examples:
Blitz: his pull is WAY faster than thresh's. his knockup is not a stun. and you're forgetting that he has huge base damages compared to many other supports and a ms buff to get right in your face with.
Leona: she is a butthole full of cc and damage boosting. also, her e is a root, not a stun, and any good Leona will land the stun on her ult more often than not.
taric doesn't really count, because he is generally agreed to be underpowered compared to literally every other support and riot is working on a rework for him.
sona: her damage is by no means weak and by no means single target. her heal may be low (ish, only a little low) but its on a short cooldown and also grants a shield.
soraka: her heal heals for a TON - yes it hurts her, but a good soraka will sustain through most of it with her q. she does not have super weak damage, as any ap soraka who knows what shes doing will gladly show you. her silence is aoe, not target (and it also roots at the end) and her heal isn't aoe, it's basically a reverse karthus ult (auto targets allies and heals them).
name: ok, that's not target, its an aoe skillshot. and its a knockup, not a stun. the buff is not to ad, it's to damage and it gives a slow on aa's. her ult, again, is not a stun, it's just a knockup and it has a massive range.
you're right, you really don't need to go on. a lot of your facts were circumstantial, false, and/or had missing key information. utility is not the only thing you want in a support, as Leona's passive and braums shield prove very strongly. supports are about securing kills and keeping your allies alive, not just literally moving pieces around. thresh is a jack of all trades support. you know how in all the fighter games you have guys who are especially strong, or fast, or difficult to kill, and then there's the one guy who's average at everything? that's thresh.
besides, if utility were as dominant over everything else in a support as you say, zilean would be way more popular right now.