No, your 2024 1099-SA should look like this:
Box 1: 2887 (the original excess)
Box 2: 1,590 (but only if the HSA custodian grew up and calculated the excess the same way that you did)
Box 3: 2 (indicates a withdrawal for excess contributions)
Box 4-6: empty
"Therefore I should be withdrawing $4,477 from the HSA immediately, with reason of "excess contribution"." - This is backwards, but only because your HSA custodian refuses to do their job and calculate the earnings themselves. The way it should work is that you ask for the withdrawal of the original excess, then the custodian calculates the earnings, then the custodian sends you a check for excess and earnings, and the custodian sends you a 1099-SA with excess in box 1, earnings in box 2, and distribution code in box 3. Simple if your custodian would just co-operate.
HOWEVER, since your custodian will not do their job, do what you said about withdrawing the 4,477, and edit 1099-SA to match - UNLESS the custodian puts something in box 2 in which case you'll have to go to plan B.
"turbo tax will prompt me about what to do with the earnings?" - the earnings are entered as part of the entry of the 1099-SA. You are not separately prompted for this.
In any case, you must document what you did and why, in case someone asks about it later, because you won't remember by then.
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