I thought you could already do this on iOS? I can’t remember where I saw it but I could’ve sworn you can without jailbreak
1) you can sign an app for your device and run it without paying Apple anything for 7 days. After 7 days you have to do it again. Not convenient in the slightest, but still possible for free.
2) you can pay Apple $99/year for a paid dev account and the 7 day window extends to 365 days.
Neither of those above options require source code, somebody else could’ve compiled the app, you just need to sign it.
3) TestFlight allows installation without full App Store review and builds expire after 60 days (or is it 90, I forget?). I believe the cap of “limited people” sibling referred to is 10,000 these days, but I could be wrong. iSH is an example of a popular app using this route.
4) Corporations can sign up for an enterprise account for $299/year. This allows signing apps similarish to option 2 above, but for a virtually unlimited number of devices. The apps are required by policy to not be distributed to the general public though.
At the end of the day, you can't just download an app and install it. You have to go through hoops, all of which involve money, expirations, or limits.
2) you said you needed source code and to compile, that’s not true.
3) you said you “can’t just go online and download an app”, which unless you’re being very pedantic, isn’t true. I even gave a popular example, iSH.
OP didn’t ask if alternative app stores existed with one click installers, they said you should be able to side load apps on iOS, which I’ve shown true via multiple methods. If you had “said exactly the same thing” I did, then you must have a different definition of the phrase “you cannot” which is the lead of your comment which I’ve shown to be factually incorrect.
But you can't just go online and download an app.