If I understand the information you provided, a loan did not exist until 8/30. Generally a contract is not consummated until the borrower signs the note. Once the loan is consummated you may begin the accrue interest, if, as you mentioned, allowed by state law.
I have seen a few transactions similar to this where parties live in different cities and can’t all be present at closing. Documents are prepared and shipped to another city to be signed, and then are shipped back to the closing agent. That might explain the delay from doc prep to consummation and from consummation to funding. Even then it does not make sense to book a loan with a date prior to closing.
If all parties to your transaction were present at closing, documents should have been rerun for closing. At this point I can’t necessarily recommend reprinting the documents. I assume the mortgage has been recorded. You have a real can of worms. I suggest having the bank’s legal counsel untangle this mess.