Can driving a car with a bad A/C compressor cause issues with the engine?

Had car repaired 01/02/24 and after payment was made I read in the report that my A/C compressor was found/noted as bad and needing repair, however, the tech did not report it to me or discuss it with me at all. It was noted that I had “declined” the repair. I was told that the tech decided that I would probably want to wait due to it being winter weather. After reading online, I see info that driving a car with bad compressor could cause issues with engine. Performance? Can this repair wait? I notice vibrations in steering wheel/rough idle and feel I may have smelled “burning plastic” when using defrost on below freezing morning. Unsure if related. I have not allowed the a/c to come on. Called them back this morning and they told me there was no concern and it was fine.