Coming from a support side:
I rarely deviate into some of the non-standard type picks, like support Shen or support MF. Most of the time, my roster rotates through Ali/Braum/Leona/Janna, and I have fun playing those against most matchups. I would like to add Brand into it, but from playing Mid Brand to Bot Brand, I haven't quite tuned my 'Oh yeah, I can totally take this fight. I'm mid Brand!' into account and I'd rather not feed the enemy team with it just yet.
Most of the times I don't have to ask my ADC if my pick is good. However, if they ask for something different or I pick someone to better align with the rest of the team (dropping Leona to play Ali or Braum if we have a Yasuo for instance), I'm more than happy to rotate into it. At the end of the day, it's a competitive game and I want to win, so playing my best and providing as much value to both my lane and my team as possible is ideal.
Coming from an ADC side:
I will admit I prefer standard picks over non-standard for two reasons:
- I know exactly how their kit operates in the lane and how to play with/around their abilities
- I know how their damage/damage mitigation can help me in lane
However, in most cases, this doesn't mean I won't play with a non-standard support.
I'm not adverse to having something different come out on the field, like support Pant or support Poppy. I usually love getting asked 'hey, this okay?' over an instant lock, but I'm not angered or disappointed by someone instantly locking in their choice. What I do care about, is how they're going to play that champion and how best it'll play with me and the team.
As a random example, there are a lot, dare I say most, people who can play support Trundle far better than what I could do in Support, Top or Jungle with him. Playing ADC with him in lane, I'd expect he'd be applying movement slows for poke/ganks, pillars for disruption or zoning, soaking some damage thanks to regeneration, and dealing % damage/stealing stats for level 6+ fights. I'd hope he'd ward, we'd play to poke, and late game he'd be ulting a tank for extra stats and making the team fight just a little easier, and if I'm really lucky getting sweet engages off of pillars. I wouldn't really expect him to want to take a lot of CS but for Targon Stacks/me not in lane, I wouldn't expect him to build AP, I wouldn't expect him to ignore warding.
But that's operating off of a pretty big assumption that goes all the way back to champion select. He got support and he'd be supporting me (the ADC) and the rest of the team (as supports usually do by various methods). The other players on the team would expect he'd be bot with me, and so on.
All that can change as soon as he/she says 'I'm support AP trundle.'
Now I know his ult will be his main source of damage, he'll probably not be super tanky, and lane may be hard if he's not got a Targon's for extra health back. I can trust they know what they're doing, or I can ask 'hey, I'm a bit unsure how well that'll go with Cait...do you have a different pick you're okay with?' But as soon as it's locked, gotta work with what's here, and trust he/she will be able to support. I can always ask before they lock, but at the end of the day, we all have to work together to win and we need to play complimentary styles to ensure everyone's on the same page.
It's that covenant of trust between adc and support that has to remain intact for it to be a successful lane. That could mean having an Ali doing wombo combos so you can't miss your Ashe R unless you literally sneeze as you're firing, to a Bard who's going to leave your lane to collect those sweet sweet chimes and hopefully leave you some health packs to top off with, to a support Pant who decides to push your lane, gank mid and then gank top before coming back to you.
Every single one is fine. As long as everyone understands what's going on and everyone's cool, absolutely no problem at all.
In closing, I like defaulting to a single phrase when someone is complaining about who they have to pick into, unsure what they want to play, or just have no idea what works best in our comp since they're first pick. Doesn't matter who's picking what, or for what role. What matters is that we all understand what each component of the team is wanting to do and tries to with their picks and plays and purchases.
"Just play your best!"
or one of its variants,
or 
, and try and not take too many of the ADC's kills if possible, especially if they are the only source of AD for the team and/or if they are mid-late game scalling (like Jinx, Tristana, Vayne, and Twitch) because they can use the full gold amount to get ahead and hit their power spike faster (snowballing in a sense).
) have been sort of pushed aside for a few dominant ones (
and
) for most in not all of the past split? Until just recently, it was rare that anyone other than maybe Jinx, Twitch, and Vayne made it into a game where one or more of the dominant champions was left open.
top, 
mid, 
jungle, and 
^_^. the play potential with him is great.