I ran into this many years ago with my suburban and wife's Tahoe
But my 03 suburban you have to have the snowflake on for ac to work and on the 04 Tahoe turning it on, will turn the ac off. If you look closely you'll see a line through the snow flake where it means off
And if I remember correctly cause it has been many years lol. Something about the early models/ software automatically had ac off and you had to command it on. But then they changed it too ac was always on with the control and you had to command it off instead if you didn't want it.
I also recall if you buy a replacement control that they the same with ac on always and you command it off
What you explain seems right to me.
Also... I believe the Body Control Module (BCM) monitors and controls the AC, so swapping the HVAC control module in the dash to an aftermarket replacement would need a recalibration but should function the same as an original controller.
I was going to add to post above, that I never actually touch the AC "snowflake" button. I basically run on AUTO, based on desired temperature, all the time.
With the exception of changing the split flow to defrost/defog windshield. Never used the snowflake button, so I guess I forgot what it actually did.
Other than that, I am letting it do its own thing...
I have over 212k miles on an original AC system and it still blows cold. Maybe that system where it stays on to exercise as long as it is above freezing outside worked too good!
Maybe tis is what engineers were thinking after seeing our compressors last so long:
You sell fewer replacement compressors last that long, so let's put it back to leaving compressor off all the time and let the seals have a chance to harden and dry out... LOL