As George says, the script adds the user to model and master as well. And it will probably will produce an error when it attempts to add the user to msdb again. model solves your concern, but I am not sure that I like it.

But there is no other permission lower than CONTROL SERVER that does this on server level - unless you enable the guest user, and that's even worse. The issue is that if you access a database, you must be someone in that database. And beside being your own user, the only two alternatives are dbo and guest.

Beware that since they are oldfashioned and use sp_adduser, the script also creates a schema called AppInsightUser, and AppInsightUser is the owner of that schema.