Yes you are right you have to remove Built-in Admin account from SQL Server. In fact Microsoft advises to remove it to tighten the security. A few months back we opened a ticket with MS and they guided us how to do that.

BUT BEFORE YOU DO THAT......

Make sure you have SA account password and fully tested. Or any other account that have Sys Admin rights. If you as a DBA accesses the databases using your domain account, make sure you make your account or the group (if you are using a group account) has sys admin rights to the server.

As always, test these changes in dev enviroment before you do that in prod.