My car overheated and then shortly shut down on its own

The thermostat is the last thing to think about. The leak at the hose would have introduced air into the cooling system, and that can cause the thermostat to close leading to overheating if the coolant level falls low enough. Thermostats open in response to hot liquid, not hot air.

If you didn't drive too far with the overheated engine, it is most likely no permanent damage was done, but it could be necessary to burp the air out of the system once the leak is fixed and the system is refilled. Some engines do that on their own. Some have a bleeder screw to open or a sensor or plug that can be removed to let the air out while filling the system.

Fix the leak, fill the coolant, then see how the engine runs. If it overheats again in about ten minutes, try burping the air out of the system.

BMW is one of the three most customer-unfriendly manufacturers. Unfortunately you are stuck going back to the dealer for many repairs but any independent shop should be able to check the engine for you. Overheating is never good, but you didn't drive far enough to CAUSE further damage, and while it is odd the engine shut off, that may be not related to the overheating issue. I would not waste money yet on a lot of diagnostics when you already know there is a leak to be fixed. Start with that.

Myronj, how goes the BMW? What happened after you fixed the leak and refilled the system?