As a manager I would only do this if I really wanted to fire the person already. For employees I care about I would give them an out.

Part of being a good manager is understanding your employees and helping them succeed. If someone makes a statement like this in the heat of frustration it doesn't necessarily mean they will actually quit. If they're a valuable member of the team you should present them with an opportunity to save face and remain.

To me this seems like taking an opportunity to fire someone they already wanted to get rid of. Either that or a bad manager who wanted to flex their power as a threat to the rest of the team.