Should the Update Manager module be uninstalled when installing with Composer?
Personally, I have Update Manager installed on all my live sites.

While I use composer to manage all of my sites, I am do not blindly composer update and push out.  I like to read changelogs and make sure I regression test thoroughly.

So, I have email notifications for updates turned on for all of my sites.  Most are set for daily notification for all updates, but a handful of internal-only sites (as in behind the firewall) set for security updates only.

Then I can test and plan what updates go out when during my maintenance windows.  Or for security updates, triage how critical they are to go out outside the normal windows.