1) Why would an Orthodontia provider want to receive this payment at
all in the first place? I was thinking maybe tax reasons.
Less cynically, medical providers request that insurance payments be sent directly to them (and you routinely agree to allow them to do so in the fine print of your insurance contract and the paperwork you will out with the provider), every time that they request payment through insurance, because it is extremely uncommon for someone to pay in full and then come up with insurance through which a claim can be submitted. It is set up this way, in part, because medical providers have a lien on insurance payments made to compensate any not previously paid for services that they provided. So, anyway, bureaucratic inertia causes it to happen this way in your quirky case where it really doesn't make sense.
Also, insurance companies routinely see claims failed by providers and handle them uncritically, while rarely seeing claims filed by insured for services that have already been rendered and paid for, and so look at those claims more critically, so it was probably in your interest to have the provider make the claim for you on your behalf.
More cynically, even if the medical providers doesn't get to keep the money, holding onto it helps their cash flow. It may allow them to borrow less money for operations, or to invest it and earn a return on it. Having money has value even if it is temporary.
2) If they never end up sending me the claim refund, do I have any
legal power to fight against this? I have all of the original receipts
and an EOP report.
Yes. You could write a demand letter, including copies of your evidence, and if they refused to comply within a reasonable time, you could sue, either with a lawyer, or in small claims court. You would have an extremely strong chance of prevailing and in all likelihood they would refund the money before the case went to trial rather than fighting a losing case in court.
You have solid evidence that they were paid twice for the same thing. You don't need to bring the insurance company into it (which would make the case much more complicated). You just need to show that your local medical provider got a double recovery and didn't provide a refund.
Six to eight weeks is unreasonable when they have already been paid twice. But, given the cost, aggravation and delay associated with suing, you might want to give them some time to provide a refund, although the next monthly billing cycle for them ought to be more than generous accommodation from you.