As far as I'm aware it's mostly a hack to avoid UAC elevation. If your goal is to get Chrome onto as many computers as possible: you have to be able to install without admin privs. (Also, even if the user can elevate themselves, I imagine the conversion rate is marginally better if you can avoid the "scary" UAC prompt. The conversion rate is definitely better if an unprivileged user can avoid invoking the wrath of their IT admins/policy.)

Right. I just feel like the opposite direction is where we want to be. I don't understand in what world is it acceptable to install applications without the user's (or operating system's) permission?

This is literally a side channel installation of software. As much as I hate the store concept, I would prefer everything be handled via a store than this mess.

Installing per user was an MSI feature that I'm pretty sure predates UAC.

The idea was not about installing without user's permission, but was entirely about allowing normal (non-admin) users to be able to install software, which would obviously be limited to their account only.

This means it needs to install to part of the filesystem where a non-admin user has write permissions. There are only two possible places. The user's profile, or the Programdata folder. Installing to the latter would have problems where one user could maliciously modify a program another installed (or issues if two users installed different versions of the same program), so the user's own profile is where it must go.

This concept is nothing new, users could long compile and install software into their home directories on unix platforms.