I have all but nine champions and played under six of them across twice as many games, so I can't say I feel you.
Random is random. It won't fix your problems. If you can't face certain champions, either A) you're not meant to really in URF or a single lane or B) you need to learn their ways. They'll still be picked, though, and they'll still suck to face.
Just because all the champs are unlocked, won't mean they're all magically getting picked. In fact, us people with high numbers of champs-- ergo less chance to get someone we dislike-- will just get rerolls faster. Then, you guys have the exact same chance of getting a scrub pick, we just have a phenomenally higher chance of rolling into something that isn't comparatively weak.
This won't fix ARAM, and once you do it it won't be undone. Of the nine champions I do not own, only J4 really disinterests me to the point of what I'd call 'dislike' with a close tie for Zed-- meanwhile, everyone in the game who dislikes Udyr, Mordekaiser or Jax to name a few unpopular picks (in my experience) will be forced with the chance to get them.
Look, this is a bad choice and removes player agency. If anything, "All Roster All Random All Mid" should be a mode of its own. ARAM is perfectly fine where it is and the niche balance touches they give to it (Thresh stax) are beautiful in their own way.
If you unlock the whole roster, everyone's just gonna god damn dodge so they don't have to play literally anyone who isn't Raka, Sona, Nid or Lux.
Please. Think of the queues. Stop this madness, stop this movement and let's forget about balancing an inherently imbalanced mode.
(All of that said, it could be a nice feature for ARURF as a featured mode, or maybe a system like 1FA where the cumulative champ pool is available)