If it was truly all random with an equal chance for every champion, then you'd think you'd see even distributions across the game or somewhat close with no major discrepancies. Well, when I actually checked the stats for games playedI was surprised......not really.....to find that champions like Garen, Lux, Ezreal, Fizz, Viegar, and Zed all had around or over 70,000 games played in the last month while Skarner, Rek'Sai, Sejuani, and Zac only had about 7,000 - 9,000 games in the last month. Not surprisingly, I don't think I need to state that this shows a clear bias towards very strong URF champs and against not very strong champions. Aka, a meta of sorts.
I totally get that these numbers are the way they are because of dodges and rerolls, but I think it's pretty good evidence to show that Riot claiming it's all random hasn't stopped a meta to form and you are going to be playing with or against the same champions frequently on average. ARURF hasn't fixed anything except occasionally forcing people to not play popular champions and have horribly one-sided games while removing the ability to ban out some anti-fun champions.
Go ahead and check out the stats for yourself https://na.op.gg/statistics/champion/
The game appears to be anything but "random". You'll see the same thing for ARAM on a side note.