Houses have been built for many years with no worry about HVAC being installed before drywall. Why is this an issue all of a sudden?

We build and remodel homes in a coastal Virginia (mixed-humid) climate zone. I don't think we've ever had an incident of mold growth on drywall or panel joints during or post-installation. That said, the indoor conditions during drywall finishing are important. You need to prevent freezing and to ensure temperature and relative humidity are within a range that allows drying of joint compound without giving rise to cracking or other defects. Fortunately the acceptable range of temperature and relative humidity is pretty broad. If your concerns are keeping you up at night, you can read technical bulletin GA-236-2017 which you can download at www.gypsum.org.

It’s best to NOT have the HVAC system running during drywall finishing because it fills the system with the resultant dust.

A separate, temporary dehumidifier sounds good in high humidity areas.