I believe it is not so much whether they have any direct legal right to the physical property, it's whether or not the HOA is able to place / has placed a lien against the title of the house. A lien in the case of an HOA represents the right to the property due to some fine or fee levied and not paid. Under most circumstances, it must be resolved before selling the property, either with the owner clearing the lien by paying it, or the future buyer paying the lien by legal agreement.I find that to be dubious. if a HOA is not a legal party to the deed they cannot stop the sale, particularly if HOA approval is not legally permitted and part of the deed restriction.
The mechanism is similar to a vehicle that has been financed where the lienholder must be paid what they are owed from financing your purchase before you as the owner is able to resell the vehicle, or when the next party (like when you're trading it in to a dealership) buys out the lien in order to acquire a clean title for the vehicle.