Beamer, I guess I should have expanded on my first reply.
That was what could technically happen.
If I am controlling when an emergency happens, reading out a passage of text from my manual that requests a pilot to land elsewhere because of external considerations would be the last thing on my mind, and I've never done it, even though at one stage I was requested to do so.
I don't work at a GA airfield, so I'll leave that aspect for others to discuss.
We do 'manage' emergencies to a certain extent. If an inbound has a hydraulic failure and is doubtful of its ability to vacate the runway, all other things being equal, we'll land it on the runway which at the time would be the worst choice for gonig single runway, leaving the better one available. All that goes out the window ifv the crew requests a particular runway.