It is depending on the company.
My past experience was monitoring 40 servers by myself(in Healthcare Industry), now ~ 300 servers all over US (in Financial service) by only 2 DBAs but mainly all maintenance job monitors by myself the other is senior DBA ( mainly in Application).
For monitor ~300 servers, it is very easy by a script and let it pages us when it has the problem.
Big problem always comes from a disk problem, need to change server and pretty much we have all backup files and some database files to recover.
Backup, Update stats,DBCC, shrink the file, restart service etc, all done by jobs by the server itself or from monitoring server.
All can be done by set one computer as monitoring server.
All update can be done automatically by the network process.
IMHO, reply to the question, How many DBAs Does it take to manage infrastructure? It should be 0 (zero) in nearly future.
No need DBA for infrastructure in the future.
(Some Company still needs Database Developer for the future).
It is possible, let Network guy install server and SQL server. Monitoring Server can detect the new server, install all jobs automatically by itself.