i would like to make this as simple as possible.
Bard/Ivern/Thresh = tier 3 supports.
Lulu/Nami/Braum/Rakan/Pyke/Zilean/Taric = tier 2 supports
Sona/Soraka/Janna/Morgana/Tahm Kench = tier 1 supports
The benefit of playing a tier 1 : simple and powerful. typically builds standard items to win. easy to counter (grievous wounds, chill, pure damage), very minor skillshots or ally interaction.
The benefit of playing a tier 3 : Unique gameplay (chimes, souls, jungling without hurting camps) , difficult to predict gameplay , flexibility in build paths and ability use (Ivern's bushes, bard R, thresh flay) , requires a lot of allied communication (leaving lane as bard, making sure someone grabs the duo buff as ivern, throwing out a line for allies as thresh) , has HUGE end game potential (unlimited soul stacking, chime stacking, free camps post 20 minutes as ivern
the benefit is you can express your skill more as a tier 3 difficulty. the downside is that you're going to have to rely a ton on your allies and enemies making the right (or wrong) plays to get to where you want to be.
i hope that helps.
to answer the rest of your questions - it depends on your end game goal and game format. you shouldn't for example be expecting to do well with team reliant champions in order to win in solo queue or low flex queue.
if you're playing with a premade that is skillful then yes you can benefit from flexing your skills and teamwork on your opponents with ivern, yasuo, azir, viktor, zoe, etc.
otherwise you're going to be waiting and waiting for your allies to make a play that you can kickstart your build from and it's probably never going to happen.
in those instances i recommend the level 1 difficulty or level 2 difficulty champs.
if you dont want to be a heal bot or a stun bot and want some interesting mechanics without losing your solo carry potential then that's what tier 2s are for.
zyra, karthus, vladimir, sivir, corki, shen, poppy, evelynn, etc.
they add a bit of expression while also being semi self reliant.