Does compressor have to run when filling the air conditioning?

The car workshop first found that there was no more refrigerant in the plant. However, the system was tight. It was also found that the air-conditioning compressor was not working. I then got an estimate of 780 EUR for the replacement of the air-conditioning compressor. I then took a look at the air-conditioning compressor and found that the clutch on the air-conditioning belt disc was defective:https://youtu. be/IXq3z68EbUk?t=106 After the coupling was changed by me, the air conditioning system is now going again! I went to the workshop again, with the request to finish the air conditioning service completely. After all, I thought that the air conditioning compressor had to run so that the air conditioning could be filled properly. The mechanic then said that they had already refilled refrigerants, and the compressor did not have to run!? What is the real thing? is the system running, or is it even possible with a broken compressor?