My experience in the three HOA communities I have lived in here near Phoenix, AZ all have had antenna restrictions in their CC&R's. I would assume that the "footprint of the property" is all inclusive of structures on your lot. At least that's how the HOA can interpret it without being specific. But the fact that you "must get approval for any structure..." leads me to believe that if you don't apply for approval and erect something (antennas, sheds, etc), the HOA can have the authority to make you remove it. Another thing to check out in the CC&R's is any provision that precludes erecting anything that can be a detriment to the health, welfare, and safety of the community. Antenna structures can be construed as part of that provision.
In my opinion, to ask approval for a tower that is visible in any way to others in the community would be a longshot. I could be wrong. Personally, I have always subscribed to the saying "It's sometimes better to ask for forgiveness than for permission". For me, it's worked out OK, using stealth antennas that no one can see (attic dipoles; ground or steel fence-mounted screwdriver), and always able to enjoy the hobby. On your half-acre, a ground-mounted screwdriver would permit a sufficient number of radials to make it very effective from 10-80 meters. I've worked over 285 countries and achieved some decent contest scores using one, but YMMV.
If you do ask for permission and they approve it, then great. If they say no, then you go the stealth route. But you have already alerted the HOA that you're a ham operator, and they can be jerks and be on the lookout for anything in your yard. Every HOA administration is different. A damned if you do, and damned if you don't situation. Good luck with your decision, and ability to get on the air.
73, Bob K7JQ