A/C does not blow cold air?

The air conditioner does not turn on.

I recently bought this vehicle and have just discovered the aircon does not blow cold air.

After turning on the cool fan to setting 3, I then press the aircon button, wait a couple of minutes but no cold air comes out.

I asked a mechanic to look at it. He added some freon and activated the aircon and said there was a loud screeching sound so he immediately turned it off, worried it might cause more damage.

I was advised that I needed a new pump. I have found a pump that I can buy online, but I am wondering if I can do some more checks to see if the pump really needs replacing. Or if there might be some other problems - such as the turn on signal not getting through to the compressor from the aircon button in the cab. Or the compressor gap not being correct.

I have looked at a couple of YouTube but am feeling a bit overwhelmed by all of the possible failure points and was hoping for a bit of help to direct me to what to look at first to better diagnose what the actual problem is.

Attached images:
- compressor
- high low valves
- aircon radiator