Switch to using the new SqlClient user-agent feature

In #35730 (EF 10), we started injecting Application Name into the SQL Server connection string, including information about the EF version.

This was a stopgap mechanism, as SqlClient is in the process of introducing a more first-class user-agent-like mechanism for submitting such information: https://github.com/dotnet/SqlClient/pull/3451, https://github.com/dotnet/SqlClient/pull/3489.

Once that's done, revert #36548 and switch to the new mechanism.