They should, displacements apply a stun on the target after they hit, so they are actually 2 cc in one, but since the 2 cc overlap each other almost perfectly (or at least when they don't it's clearly written) in the end they count as one huge cc.
The other toggles you are mentioning are not disabled no matter what, they still continue to work even through stuns (I think anivia's ult does though) suppresses, fears and charms.
The counterplay is not too much, it's just clear and like all the champs with clear strenght and weaknesses (who are always "weak" unless they have strong numbers, which sol does have) it's just that easy for more overloaded champs to actually play around it, but he's still got the tools to deal with them. Also it's Sol's job to keep you from reaching him and he even got a strong and fast displacement to play around enemies coming in for him and a huge ass aoe stun which can easily fuck up most opponents and give him more time to reposition since it's impossible to dodge his stun in melee range unless the sol player is just bad at throwing it or the enemy has some kind of mobility, but even then the enemy will still be near your strong range or if he dashes into you (bad idea, he'll probably get stunne anyway) and somehow manages to not get stunned he won't have a tool to escape from your ult which will bring him in your strong range again so he's still fucked.
I think that a good Sol can be almost like a good Azir: it's hard to pull always off but they have all the tools they need to deal with EVERYTHING in the game, I remember someone saying that an amazing Azir has no counters, and I think the same can be said about Sol, he's just got exactly what he needs to deal with everything: no less no more that's why he feels weak compared to other champs which have far more than they actually need to do theyr job.