Was unable to install any app - how to find out what was blocking the installs?
There is a low space threshold of about 500MB (depends on the device), and any app install or update that would take you over this limit always fails. No-one really explains this, least of all Android itself.

This is actually a badly-implemented "feature" - it is supposed to prevent you hitting 0% free so that system apps don't start failing, but it doesn't fully prevent that. You can still hit 0% free - and get into a terrible situation where system apps pop up error notifications every fraction of a second - by uploading/downloading files to the phone.

(The reason the 2 apps were able to reinstall themselves, is probably because I cleared some caches for some of the apps in settings as well as uninstalling those apps.)

The only way around this problem involves root access - which I am not allowed to do because it violates the terms and conditions of my work partition on my phone.

So the only thing to do is free up enough space...