MacOS – Renaming user failed, lost admin rights

I wouldn't recommend that you try to put in such a scheme. If you wish to use restrictions, you would generally not want the restricted user to install any application either (for security reasons).

Normally, a non-admin user cannot install applications in the system applications folder at /Applications since it's owned by user root and group admin. However, a non-admin user can install applications under the home directories (~/Applications), but those applications would be available only to that specific user (certain applications that are designed to be in /Applications or those using installer packages to write into system locations may not work well in such a case).

If you wish to allow a non-admin user to install applications into /Applications, then you can add that user to the admin group from Terminal.app (instead of going to System Preferences and allowing the user to be an administrator). Keep in mind that a user who can install applications into /Applications can also remove applications from there (intentionally or unintentionally).

There might be a way to modify the standard behaviour by doing some fancy low-level unixy stuff like creating the "root" account and logging in as that which would cause the authentication dialogues to not be so intrusive, but for the average user, the answer is "no".

A non-administrator account will be presented with an authentication dialogue in some situations where an administrator account would not have been presented one (such as when modifying items in the /Applications folder), so perhaps doing system updates might be easier when logged in as an administrator. You can change any non-administrator account into an administrator account by selecting the “Allow user to administer this computer” box in the “System Preferences” -> “Users & Groups” settings window.