The backups with time shift are incremental, hence most of the time the backup is taken within seconds and it only stores changes over time, something similar to git.
I used to do it exactly for that uses case, the backup was quick because there generally are not much changes outside the home directory.
I used to have Daily backups and monthly backup like 20 different dates stored in a relatively small space.
Like if my system is 30 gb then a 50 gb backup partition would store months of daily backups.
I have heard people have successfully utilised Mailcow without any issues for many years.
Personally gave it a try once but setting up ports, firewalls, virus scans, anti spams, dns felt too much effort for what I was going to use it for