Yeah I think that you should try to be good at one lane first if you want to take ranked serious and get a high rank.
I don't play ranked that much, especially now that I have Overwatch relatively new ;P but I am also only playing mid, when I play ranked (or atleast try to), my personal reason is that I don't feel confident enough on other lanes and don't have a big enough campion pool for these lanes to feel different.
Means that I think that you should have aatleast one lane that you are really comfortable with and where you have a big enough variety of champs to choose from, if you are being countered.
So I definitely think you should play more than only 1 champ, cause if that champ gets countered you are useless and it is stupid to think that you are good enough to dominate all the time, just because you are Riven main for example
But as always, in the end you decide how you want to play a game (not only league) and as long as you don't break some fixed rules (and are okay if you are being flamed when breaking non-fixed rules ;P) everything is allowed.
PS: Don't know if I missed the point of the discussion, am a bit tried already