Yea I think displaying enemies ELO (Skill level) will cause huge amount of drops, there has to be a better way.
Why not implement a system similar to the one in CS:GO competitive mode, where you can see the rank of your opponents displayed immediately after the game... it would prevent drops but allow you to know your opponents rank/ELO after the match without unnecessarily having to search for it.
I think a system were you would have both a "level- up over time" and a displayed ELO/rank would be the best solution. It rewards those who need or like the constant "success" of leveling, but on the other hand it also satisfies those who want to see their 'true' skill level and want the competitiveness of up- and downranking.
Dota2 is one example of a game that has such a system, with both levels and a separate matchmaking rank. Would be really cool if Relic would implement something similar!