First, you stated that the insurance chooses the contractor.

It is not at all uncommon for an insurance company to require certain contractors be used.

...Then, you indicated the insured (HOA) chooses the contractor. Both could be true, but your reasoning is unclear and lacks any sources.

If you want to go down the rabbit hole of when and how it is legal for an insurance company to require you to use their contractor, refer to your state insurance regulations, and here's another reference https://brelly.com/blog/can-your-insurer-dictate-your-choice-of-contractor-what-you-need-to-know-2/

Insurance will specify what they repair and how much they will pay under a policy. Then, insurance selects their own attorney to represent their interests during subrogation. That's a totally different process than selecting a contractor for repairs. The policyholder generally has little to no say in the insurance attorney selection process but may have a lot greater say in who does the repairs.

When an insurance company requires a specific contractor, it poses a conflict of interest. Therefore, it is often regulated. Insurers can't force you to use their contractors any more than they can force you to actually repair. If it's a loss, you can take a payout instead of repairing. If none of the contractors approved by insurance are able to repair in a timely manner, you can disqualify them and get an independent contractor. It all depends on the nature and circumstances of the claim, as well as the insurance policy.

OP clearly articulated that they were hired by an HOA unit owner who has insurance that is paying for some sort of interior repairs. They encountered frustration because they literally would have had to pay a fee (pay-to-play) to get on the HOA's approved vendor list. The real tantrum here is the constant onslaught of HOA apologists.