"Does a gain/loss calculation also need to be done for 2021 then?"
No. I've edited my previous post to clarify that the gain/loss calculation must always be done for a return of contribution before the due date of the tax return, which is what you were asking about in regard to the 2023 excess contributions. The corrective distribution for the 2021 excess is being done after the due date of the tax return and must be done as a regular taxable distribution of exactly $475.01 as I previously described.
Regular taxable distributions are subject to ordinary income tax, meaning that the money would be taxed twice since the original excess contribution was not excludible from income, and to a 20% extra tax since you are under age 65. With a 2022 excess of $3,600 and a 2023 excess of $1,800, correcting these after the due dates of the tax returns would mean unnecessarily paying $324 in excess contribution penalties, $1,080 in early-distribution penalties and, say , $1,1188 in income tax if this falls in in the 22% tax bracket.
"they can't give me the gain/loss calculations, since the HSA funds were transferred to a 3rd party broker (TD Ameritrade) for investing, and they have no insights into the activity in that account." That seems like a lame excuse, but it seems that it would be a waste of time to argue that. My own HSA is similarly invested in a mutual fund, but the HSA custodian somehow always knows the daily value, so it has the information needed to do the gain/loss calculation (although I've ever had to ask them to do the calculation).
My guess is that you'll need to do your own research to determine the adjusted opening balance and adjusted closing balance for the gain/loss calculation described in CFR 1.408-11. If information is unavailable, you'll probably need to make your best estimate. The IRS is unlikely to question the calculation, particularly if the HSA custodian properly prepares Form 1099-R using the gain or loss value that you provide to them for the 2022 excess and the 2023 excess.
