If I understand correctly, DNS propagates everywhere anyway so what difference does it make? Or are the DNS records not cached/replicated?
It does, and they do. Propagation, caching, etc. all happens with DNS. But that doesn't mean that you don't still want your master DNS to be screaming fast, globally distributed, highly available, etc. If your DNS goes down, most DNS providers (Google, etc.) will known almost immediately and see your infrastructure as offline as part of their security system.
Does Let's Encrypt give SAN certs? I was under the impression that it's for single one off type situations where people don't have the cash to purchase their own cert, most likely being used by someone self hosting (or single site/server hosting).
If it is free, there is no reason not to get 4 or 5 certs instead of a SAN for most things.
Stupid question time (cause I don't know) can you install multiple certs on the same server?
I don't know about apache, but with NGINX each virtual host can have it's own.