This is possible to do, but lots of engineering. You can provide the experience of a single DB while each tenant can be placed in their own dedicated Postgres compute. This would help the application to stay the same while tenants are moved to independent computes (you can even move only a few tenants and leave the rest on a shared Postgres compute).

What do you mean by "rewriting everything"? Or maybe your definition of "per-tenant database" is different from mine. In our product, it's just a small layer which routes requests to the target organization's DB. When an organization is created, we create a new DB. Most of application code has no idea there are different DBs under the hood.

There are logical DBs (folders/files on a single physical server), and there's a few physical servers. We're currently at the stage where the first physical server hosts most of smaller organizations, and all other physical servers are usually dedicated servers for larger clients with high loads.