I suspect you're not calculating odds correctly, and/or making some incorrect assumptions there. (For example, it seems likely you're implicitly assuming that all 10 people in the match own all champs, which is almost certainly false.)
Accounting for rerolls also makes determining odds tricksy.
Rioters have occasionally chimed in on ARAM randomness (maybe it was in an ask riot or a /dev blog or something, or maybe posts I don't remember), and they consistently say they've looked at the code and it is random, it's just theres a lot of confounding factors that make the champs you ultimately see see less random (not to mention the human tendency to find patterns even in random data).
For example, some champions are just way more popular than others (e.g. jinx/lux) independent of their power in ARAM, so a lot more people are going to own those champions than own someone like udyr. Those champs also happen to be strong in aram, so if someone gets them (somewhat likely given most players will own them), they're likely to keep them. By contrast if someone gets a bottom tier aram champion, theres a pretty good chance someone will reroll them, so you're likely to see strong aram champs more so than not so strong ones.
On top of that, dodges are pretty common, so if a team comp doesn't seem strong, it's not uncommon for someone to dodge, so the champs that actually see play on the abyss tends to be biased towards champs that are strong there, despite those champs being weighted exactly the same as far as the code randomly picking champs is concerned.
Free week champs are also naturally going to show up more, since those are the only champs everyone is guaranteed to own. Both zilean and jinx are free week currently, so it's no surprise you see them often enough. Thresh is one of the most popular supports, so also prone to showing up. Corki is somewhat popular too, and it's only really illaoi that seems kindve like an oddball, but with enough games played and the other 4 being somewhat common, it's suddenly far more probable than you'd face the team comp you describe than the multiple lightning strike scenario you're toting.
If they want to make the aram experience really feel random, they'd have to unlocked the entire champ pool for everyone there, and remove champ rerolls, and prevent dodges from rerolling your champ. And even that would probably not necessarily feel random to everyone, since people are bad at recognising randomness.
tl;dr: It's uniformly random across a players available champions (not all champions), so it's "random", but what champions a given player has available is not random, which naturally biases peoples initial comps towards popular champs (many of which happen to be strong aram champs), and then champion rerolls bias that already biased result towards comps that are strong in ARAM.
