I don't mean one database server for each customer. I say one database for customer. Hundreds or thousands of customer can be on the same database server. When you need more resources, you add another server. If customer grows too much, you move it on another server.

There is a bit of overhead, but not huge by any means.

Being able to update schema one customer at a time is a huge plus in my view, as it gives you a lot of flexibility in rollout. You deploy a new version of the application on a new application server and move the customers on the new servers updating their schema one by one (automatically, obviously)

Backups are routinely automated anyway.