I get what you're saying, wish I had more specific control over the compressor. However, this is not just a Mazda or Mazda 5 thing. Pretty much every single newer car with auto climate control will use the compressor when you've set it to auto. Whether heating or cooling, it will always use it to de-humidify the air in order to reach and stay at your desired temp faster.
Then, yes it makes sense when you press the actual A/C button that nothing happens... because the A/C is already on. When you press it again, understandably, the system thinks you want to adjust climate settings yourself, turns off the A/C, the compressor (depending on where you're sending the air) and the Auto Climate Control.
Some FYI's just in case you didn't know... 1. When using any form of defrost mode or mixed defrost and cabin air modes, the compressor will be used. 2. When sending air to just the footwells, the compressor will be used. In both instances, the A/C light does not come on... because, again, it may be being used for heating rather than cooling. Yes, it can be confusing... for some... but once you figure it out, it works well and makes sense.
I wounldn't worry about the compressor... I live on west coast (wet coast) of Canada and pretty much have to run the compressor constantly all year round in order to keep the windows clear... as does everyone else over here. Compressor seems to handle the load just fine.
However, again, I do wish I had full control over the compressor. I feel I should be able to tell the Auto system to heat or cool my cabin to my desired temp, and if I feel I need to dehumidify, I should be able to turn this on or off.
When I used to have a Protege5 back in the day... in the forums here, someone had posted a "How To"... I don't remember exactly how it worked, but they installed a work-around giving you greater control of the compressor. Of course that car didn't have auto climate control, so it just didn't allow the compressor to kick on during defrost or footwell only air unless you physically turned on the a/c. This is how I think all cars climate control systems should be engineered. I guess car companies think we're too stupid to be able to handle these difficult tasks ourselves. Yet in some cars they allow for many variations of traction control settings!