Q&A

The fan motor is not the same as a fan clutch. Two entirely different things.
The fan clutch must be locked up and is therefore driving the fan too hard. Your engine will not overheat for that reason because the fan's running too hard, not failing to run and then overheating the engine.
Replacing the fan clutch is not a time consuming or difficult job to do so it's the parts that are expensive. The difference between an adequate quality fan clutch (thermostatically controlled) and one that's not good enough (viscous fluid drive) for some applications is not just the $150 or so difference but that the clutch doesn't drive the fan well enough in high heat conditions. It works on a simpler system and is therefore cheaper.
When I hear a locked up fan clutch roaring I always worry about something coming apart and flying through the hood but that probably wouldn't happen.
Naturally, I have to recommend replacing the failed clutch because I'm a mechanic and that's what I'd do for myself, of course, I don't have to pay labor and I could get a good deal on the part.
You didn't say which engine you have or what your driving needs are. If you are satisfied that you can wait to replace the fan clutch, I don't think there'd be any bad effects unless it's that there is a greater load on the water pump bearings due to the increased fan speed.
Here's a directory link for you if you want to find a GM specialist, maybe shop for a low labor price rather than a cheaper fanclutch:
https://repairpal.com/directory?address=30510&car_bra...

Fan clutch is designed to lock in when is hot enough to actuate. That is that jet noise from failed clutch that is locked in all the time. Rad fan motor is not the same thing. Us repairpal to help you decide on correct estimate price.