The two solutions you offer, while good are slightly unrealistic/hard to pull off and don't really stop ARAM accounts.
Your first solution is for players with larger champion pools to ban a percentage of their own champions to ban away. There are a few problems with that.
First, it still doesn't make it all random, because you're still banning away champions/making a pool of more favourable picks.
Second, how would that even be implemented? It would take a lot of time for each person in champ select to go through their own roster and ban away certain champions that they don't want to be given. That would make champ select a lot longer as well.
Third, that doesn't stop ARAM accounts from getting away with owning favourable champions.
Your second solution is a better idea, but still can be manipulated. How many champions would an account need before being eligible to play ARAM? 30, like ranked? Well lets see if there are 30 favourable champions that an ARAM account can have:

Well okay, 40 then!

50?

There are few more good picks, but these are some of the best (from my own personal bias and standards. if you really want the best champs for ARAM I would suggest looking at a Tier List)...
But requiring 50 is too much to ask of people, but if so, there are plenty of viable options that ARAM only accounts can still manipulate.
So Riots best options are either allowing bans or making every champion available in ARAM. If they choose the latter, then having a pool of 143 (I believe that's how many champions are in the game currently) are gonna make a lot of people unhappy when they really do get a champion they don't know how to play meaning, you're gonna see an increase in dodges. And a lot of unhappy campers, grumbling about how they didn't get a better champion.
So currently the former, allowing bans seems like the best option and again it is experimental.