Well, it's random within the confines of all the other criteria it need to fill, but theres a lot of mixing factors that can feel pretty bad for those amongst us who have most champions.
Plenty of people have relatively few champions, and the ones they do have tend to be ones that are fairly strong in aram. Not just ARAM only accounts, but stuff like lux and ashe are pretty common early champ pick ups for even regular rift players (plus many newer players tend to prefer ranged champs over melee anyway, which also bias the odds of them getting champs that happen to be good in aram). So the result is often that the 'good' champions are played by people who have smaller pools.
Rerolls and dodging are also pretty common in aram, so often times if a comp doesn't look strong, at least one of the 5 players will be willing to use reroll(s), making the initial meh team look potentially better. If not, it's common for someone to dodge. If it does look good, then you have to hope the enemy team doesn't have someone dodge (which again depends on them having a decent looking comp). Rerolling if you only have a few champs is that much more likely to produce a 'good' champ compared to if you have 100+, so once again the small pool people are likely to get the good stuff, leaving meh stuff for the large pool players.
Also, large pool players tend to have more overall game experience (as after all, they've played enough to earn ip for those champs), so are more likely to be satisfied with their B tier champions since they know how to play them at least, and don't want to risk rerolling and getting an even worse one. This results in more 'average' looking comps instead of lux/ziggs/varus/etc, but since you know how to do it, you'll stick with it. Smaller pool players if they get something not so good, they may not even know how to remotely play it and immediately feel compelled to dodge instead of trying to make it work.
The solution is of course to unlock all champions for that mode, but that does come at the cost of making it a far less pleasant experience for a subset of players who may seriously not enjoy some/most champs. Is the tradeoff of potential fairness really worth it, especially considering even with true randomness that the outcome might still be determined more by which side happens to have the more favourable comp than the players' skills?
