How is the list of issued certificate templates built?

Question

Windows Server

Hello.
I have a windows server (2008 and 2012, bound to a 2003 domain) that has certificate services installed.
Tell me how the list of certificates is built that the user can get in the web interface of the certificate service (at https://localhost/certsrv) The
question is - I need a certificate for the remote desktop server (for the gateway), but there is one or only self-signed, or import from a file. That is, you need to somehow create it, but I can’t do it through the web.
In the Certificate Services snap-in, I see a dozen templates, some of which appear in a list in the web interface.
There are more of these templates in certificate template management, but still, for example, the required template Server Authentication (1.3.6.1.5.5.7.3.1) is missing, although it is in the list of object identifiers.
I'm a member of the Domain Admins group, but for some reason the templates don't appear in the list. If I add myself (the user) in the template management and give the rights "issuance" and the rest, then the element appears in the list. That is, for some reason the service does not respond to the group of administrators that is in this template with the same rights (or I don’t understand something).
UPD: Somewhat confusing. In general
1. How is the list built, what rights affect it?
2. How to add server authentication?