"the HOA reviewed the covenants for Carpenter Village, which state that approval is needed for "free standing flagpoles."" free·stand·ing ˌfrēˈstandiNG/ adjective not supported by another structure. not relying on or linked to anything else; independent.
We had a very similar case here years ago. The guy had flag pole in his yard. That's how they got him. It looks like this HOA will lose this one if that picture is accurate. That is not a "free standing" flag pole. And BTW. If they don't make those guys down the same street remove their US flags, this HOA is opening itself up to a law suit.
These HOA cases are poison for everyone involved. They waste a lot of money on lawyers and neighbors are upset at each other. If the photo is accurate and the only prohibition is on freestanding flagpoles, it is hard to see how the HOA prevails here, particularly if they are allowing other flags. Judges these days seem to feel free to substitute their personal preferences for legal reasoning however, so who knows.