alistar will always be tanky with his ult (in fact that's kinda the reason he still sees competitive play). And his cc is pretty valuable and will be the same regardless of "fed".
bard ult (and stun) is a gamechanger and is valuable regardless of how far ahead/behind he is.
braum is probably a bit squishier if behind, but he still get's good damage mitigation (I think it's 30%?) with unbreakable (and of course the 100% immunity on the first hit), and his peel with ult/q/wall is pretty solid regardless of "how fed"
janna is at never going to exceed more than 150 ap even when ahead, and getting that much ap is not mandatory. The worst you can be "behind" by with this champ is not having an aegis (which is a dirt cheap 1500 gold item). This champion is going to "win for her team" the same if she's ahead or not.
Leona has 3 excellent cc effects that don't scale at all (and are the whole reason you play her), the only way she falls behind is in tankiness, but even then, the idea is you stun the squishies and they die, if the squishy is still alive at the end of the cc chain, it's your carry's fault for not doing damage
Taric could literally have his q and E deleted from the game (not replaced, but just straight up "gone", unusable) and he would still swing teamfights with W and ult.
tahm is picked because his W is insane at saving lives. And he's up there with braum/alistar for just having strong inherent tankiness that isn't item-dependant. His W makes him a unique pick and again, just like bard ult, is completely non-level/item dependant.
Sona is probably the only support where being behind feels useless. Her best build (according to champion.gg) is actually an offensive harass build with little utility (with item's like luden's/lichbane). But that's because her rework sucks and also because her design as a champion (namely the mechanics of Q and how it harasses people pre-heimerdinger-rework with the rockets) makes her kit a lot more gimped than what it should be (and this terrible kit gimping exists in other places too, like in Olaf and with his ult). She still has an aoe stun which can swing games.
Soraka , just like Janna doesn't need ap to be good. Her most popular build on champion.gg is the really dumb warmogs tank build (which has NO ap), but if you build her as a dedicated support (aka just follow the recommended build guide, which just like janna, doesn't exceed about 100 ap, and again having ap isn't mandatory), at no point does your healing really suck if you play her well (as in not-dying every teamfight).
Thresh is in the same spot Leona/tahm is in; dark passage is unique and will always keep him relevant the same way tahm kench devour is unique, his engage is probably the best out of all supports too (although one could argue bard ult is superior as initiation).
and even though you didn't list'em
Nami, is actually pretty cc heavy if you look at her. Aqua prison, tidal wave and tidecaller's blessing are all cc effects that are valued for their cc. Her heal is the only thing that scales (haha pun), and for the most part, it's not super important beyond pushing a carry just out of lethal range from an exhausted assassin before you cc them
Lulu ult and w are why she sees so much play at the competitive level, the importance of these abiltiies doesn't change with gold
Zyra, Brand, Vel'koz and Annie are champs I'm clumping together- they're all picked for base damage, they can all 100-0 without items. That's why they see support play in the first place. Because they don't need items, or levels.
Karma is a bit more item dependant than most supports, but quite often the winning move is spamming inspire on your carry for the movespeed/shields than mantra-soul flaring the enemy.
Morgana binding and blackshield are why she is picked, binding is all about the 3 second root, blackshield for the cc immunity. Having more ap might help blackshield, but typically the shield is used to counter enemy support cc which won't pop the blackshield.
nunu isn't really balanced by Riot as a true support, neither is poppy or shen or Taliyah or Sion or Teemo or Heimerdinger or Nidalee. I wouldn't really put any of them in the traditional support list.
nautilus is picked for his cc. even when behind his ult is still "guaranteed knock up on adc for 1 second", and he's got pretty reliable engage thanks to the wideness of his hook. Also not really a sponsored support champ afaik.
Zilean has a huge aoe stun (which is the glorious part of picking him support nowadays), and the base value on chronoshift is huge and will practically always revive the player with more health than guardian angel.
I haven't really seen much of veigar support, or his new kit so I can't comment on it. I will say however, his stun is pretty strong (even when not support), in that, the ring lasts for 3 seconds which means you can't walk forward for 3 seconds (or get stunned) which is a very big deal in teamfights. And again, he's not really a traditional support, I'd clump him with poppy/nunu.
Of course if your teammates are behind, typically you will also feel behind as a support. But the issue there is perhaps your carry or assassin not dealing enough damage (since you aren't expected to carry the fight). But supports themselves are designed to be fairly item and level independent. It's why they don't last hit in the laning phase. It's why they have to split xp with the adc (or roam, which is very bad for leveling up). It's why they are often expected to give kills to teammates. You can do all of that as a support and still swing teamfights. It's just how their kits work. I honestly think OP is saying the most dumbest thing as pretty much all supports can be relevant even when behind, since it has kind of their design to work well without gold/levels (and it's also why I love playing supports, cause it's actually really easy to stay relevant; all you need to do is make good plays in the moment). Out of all 5 champion roles, they are hands down the least feast or famine role there is, the only time they are irrelevant is if their teammates are immensely behind or just plain dead.