The behavior you're observing with the Npgsql driver is expected. The Minimum Pool Size setting in the connection string specifies the minimum number of connections that the pool will maintain, but it does not force the pool to open these connections immediately at application startup. See more in https://www.cockroachlabs.com/docs/stable/connection-pooling.

To achieve the behavior you want, you might have to manually open connections up to the minimum pool size at application startup.