SQL Database Server Recovery and connections
Yes, your plan will work.  There are no hidden dependencies that will cause connections to fail.  The client doesn't know anything about the SIDs of the server or server logins.  Whatever the client might learn about the server is not retained from one session to the next.

If you want a bit of a smoother transition, you can run a script on the old server to create scripts that will create the logins on the new server with the same SIDs as the old server.  If you do this, you won't have to remap the database users to the server logins after restoring the databases.  See Sync users and logins.