I can see Epic starting a mobile game store. And, honestly, I don't see a problem with that. Apple is using Apple Arcade as one of their many "stay with us and you get all of this stuff" services, so why not? Perhaps Microsoft will join the fight and take Game Pass to mobile phones. May the best app store win.
Apple clearly has the upper hand no matter how this plays out. They'll actually have to compete, which means they may need to fix the review situation and loosen some of their policies to allow for things users want rather than make their users comply with what they want. They can stick to their "no video game streaming, no emulators, no links to your website" policy if they want to, but that'll only work as long as their users agree with them.
The nightmare scenario, for the "please don't change it" side, is that one of a handful of big players does this but also uses it to push an entire alternative store. Facebook and Epic are the most likely to try. Maybe Google, outside chance Microsoft tries. I doubt any would bother to just push sideloading an app, without using it to make a play to become the second of an app store duopoly on the platform. Basically the same strategy Steam used to gain market share—"here's our new game that everyone wants... oh, by the way, to use it you have to install our store".
I mean, Epic's been burning stacks of Benjamins for at least a couple years giving away tons of games (most of it not even cheap shovelware!) to buy marketshare for desktop app store—it's not implausible they'd give away a lot of free games or other software to get iOS users to install their store, to make a play for the #2 (or #1!) app store position on the platform. And Facebook could probably get people to install theirs by just adding some visual bling to users' posts when they use the version from Facebook's store—might not even have to give them actual extra features to make it happen.
Maybe none of that happens, sure, but it might. If these changes brought by regulations end up making sideloading easier than it currently is on Android, on both platforms, I'd say it's a certainty that something like that'll go down, at least on iOS (more money to be made there; store's more restrictive) and maybe on Android, too.