I have a tangential question. Would it not make sense for an OS, in this case Android, to have some proper mechanism for installing apps (in this case APKs) directly from a website (as lots of people have been doing fastidiously from signal.org by necessity)?

After all, this is all about trust. With software, assuming that you trust the developer, the goal is to be sure that nobody interfered with the developer’s compiled software - and who better to guarantee that than the developer themself, at their own domain? DNS resolution is already based on the “web of trust” principle, which is why you can trust your bank’s website. Arguably F-Droid performs a valuable role as a curator and selector of good software, but is there any good technical need for it to actually distribute the software?