No hard national statistics/evidence here, so please don't jump down my throat. Just relating my own experience having lived in three Phoenix, AZ area CC&R/HOA communities in the last 27 years.
My first home I had built in 1992 was purchased when the housing market was in the toilet, and builders were scrounging for customers. They let me erect a Rohn tower and 3 element tribander attached to the 2 story house, not to exceed 6 feet above the peak of the roof. If business was brisk, I doubt if they would have made that concession. Despite the approval, some of my neighbors weren't too choked up about its appearance.
Next new home built in 1997, the CC&R's stated that no antennas were allowed unless approved by the Architectural Committee. Without asking for approval, I put up a 24 foot multi band vertical with built-in radial spokes. The neighbors didn't care, but I received a letter from the HOA attorney demanding I remove it. Responding to their letter, I referenced another resident that had a pole on his house, similar to the height of my vertical, with one of those diamond-shaped wireless internet receiving antennas on top. I wrote that I would remove my antenna when that antenna was also removed, citing discrimination. I then received a letter allowing me to keep my antenna. Perhaps the other resident had approval, and they had to honor my installation as well.
The next new home built in 2002, the CC&R's had the same "unless approved by the Architectural Committee" language. This time I asked for approval, showing all the plans, for another 24 foot vertical. I received a terse "Cease and Desist" letter letter from the HOA. Still in the same house 17 years later, and enjoying the hobby with attic dipoles and a stealth ground- mounted screwdriver antenna that no one can see.
Let me point out that I went into these communities fully knowing about the antenna restrictions. Why do I keep moving into HOA communities? If you want a new(er) home out here, and don't have the budget for a custom one on your own parcel of land (and even they can have CC&R's), there's basically no other choice. Also, having a tower and beam is not my number one priority for living conditions. And besides antenna restrictions, I've had no other problems with the HOA's or their CC&R's. I enjoy the community I now live in. The HOA boards are only bound to enforce the CC&R's for the benefit of the entire community in regards to the safety, health and welfare, and overall aesthetics. Can't blame them for their decisions.
These are just three different scenarios that I've experienced. Your experiences might vary, and I don't advocate breaking the rules. I took the path of "it's sometimes better to ask for forgiveness than for permission" route. You can choose whatever you want, and see what happens. We're not talking life and death situations here. If you're told no, go to plan B...whatever you choose.
73, Bob K7JQ