What do supports do for the team?

That depends on the champion and the flow of the game. Generally their first priority is the safety and well-being of their carries, but if that is garen-teed, or if they play a champion whose main purpose is to engage, they will use their CC offensively to set up team fights with their team. Certain champions (mostly Janna tbh lol) are not very well suited for engaging and will focus solely on peeling and keeping threats off the most valuable people on their team, others are mostly straight engage like Leona and Annie, some like Nami and Lulu are versatile and will do both depending.

Generally:

Alistar/Leona/Annie/Blitzcrank = pure engage (can use CC defensively to peel but are at their most effective when using it offensively) Braum/Bard/Thresh = can do both (strong offensive AND defensive potential) Sona/Nami/Lulu = are at their best when supporting an engage (excellent follow-up but not super duper reliable CC to hard engage out of the blue) Janna = pure disengage (best disengage support you can get) Soraka = sustain/kite (neither good at engaging or truly disengaging, but will keep her team going)

They are the babysitters to those young low levels. They care and watch for them as well as keep them in line. They then turn into the team bodyguard, keeping watch for everyone. Near the end, the team appreciates what the support has done and in return use that feed to win. The support is the spirit animal of the team.

Other than warding. Depends on the individual support. Rules are... -Ward -Protect your carry -Try to mess with the other carry (zone them off farm, poke, etc...)

Being a support just entails that you can help your adc secure good farm and you keep your wards down. The rest is all your champion's role.

It's all dependent on who you play. Nami? Counter-engage. Peel. Usually helping to kite. But her kit isn't bad for helping people secure kills. Try to force fights in the jungle for maximum effectiveness on CC. Blitz, Braum, Thresh. Are all really good at zoning the enemy adc away from farm and making picks.

Heal, peel and kill steal.

But in all seriousness they provide utility to their teams. Engage, disengage, heals, Crowd Controls, repositioning spells, active ability items, cleanses, saving lives, and ultimately trying to put allies in advantageous situations.

Specify which support. They usually provide different things. Some heal, some are tanks, most have a lot of CC, some have unique utility like Bard or Tahm, etc. They all generally have many ways they can help their team's carries succeed.

they are supposed to be smexy as hell have you not seen or or even

engaging (cc), making the team fed, and protecting the adc and others

A good support will help the ADC get ahead in lane, peel for them when dove, roam the map to help other lanes, assist in objective control, CC enemies with abilities or trinkets, and buff allies with abilities or trinkets. So basically help the team win games.

An adc would straight lose any 1v1 without a support. If support wasn't a thing then you would have one lane always behind the others because two people are competing for CS, or 2 jugglers where not could go an adc and destroy any adc. Supports literally carry the carry on their back till their one job is to keep that player alive.