The thing that gets installed, if it is an executable, usually also has permissions to do scary things. Why is the installation process so scrutinized?

I think there's a fundamental psychological reason for this - people want to feel like some ritual has been performed that makes at least some level of superficial sense, after which they don't have to worry.

You see this in all the obvious examples of physical security.

In the case of software it's the installation that's the ritual I guess. Complete trust must be conferred in the software itself by definition, so people just feel better knowing for near certain that the software installed is indeed 'the software itself'.

It would raise the same kind of alert for me if someone used wget to download a binary executable instead of a shell script.

The issue is not the specific form in which code is executed on your machine, but rather who is allowed by you to run code on your computer.

I don't trust arbitrary websites from the Internet, especially when they are not cryptographically protected against malicious tampering.

However, I do trust, for instance, the Debian maintainers, as I believe they have thoroughly vetted and tested the executables they distribute, with a cryptographic signature, to millions of users worldwide.