Bad water pump?

Yesterday my '08 535 overheated (40k mi). First I noticed that there was no heat through air vents, then the radiator fan turned on very high, then the overheat.

I've read a bunch about replacing the thermostat and water pump together, but I'm just trying to diagnose whether the pump is bad and I need to spend the $400 on that part.

I initiated the water pump bleed procedure, and the pump turns on and circulates some coolant, but it sounds very weak to me.

Please take a listen/look at my video and let me know if you think if the pump is working properly.




Thanks

Mine sounds exactly like yours. I am having it replaced along with the thermostat, on Monday. Parts and labor by an indy $800. My car has close to 36K miles on it. I noticed the noisy pump after replacing the coolant and during the bleeding procedure. Luckily noticed it before an overheating incident.
I would not start the car again unless pump and thermostat is replaced.

Thanks for the response. I just bought the parts ($500) online, so it sounds like $800 all in at your indie is a good deal.