Both Discord and Battle.net are examples of services that would allow you to use _whatever_ name you wanted to use, even if somebody else has it, by giving you an additional identification number at the end of your account name that isn't visible in-game.
Does it make it slightly harder to add someone without having them already on a recent players list? Sure.
Is there any other downside? _Nope_, besides maybe impersonating a streamer(?), but even that you can usually just tell by champion choice, rank, gameplay style, or other tells if you've watched that streamer at all, and it's not like you can't just make certain names unavailable or unallowed (such as Rioter names).
Riot, if you were going to make a service that required players to completely change their account name anyway, there's absolutely no logical reason you shouldn't have just implemented this system. Two other popular services already use it because it's just infinitely better. This whole fiasco could have been avoided if you just had the awareness to see that your approach was just... stupid. Let's not mince words, it was just absolutely _stupid._