Yes, it is okay to pay the entire medical bill with the HSA before insurance covers it. However, when you are reimbursed by insurance, you do need to return that money via the HSA Mistaken Distribution.
When you pay the medical bill, you don't know for sure that you will be reimbursed by insurance, or how much you will be reimbursed. You simply believe that insurance will reimburse you in the future. If you paid the medical bill, it is a qualified medical expense that can be paid/reimbursed through your HSA.
However, once that money does come from the insurance company, that portion of the medical expense is no longer a qualified medical expense, making that portion of the HSA distribution mistaken. It needs to be sent back to the HSA.
Doing this would not result in any difference in income tax. Remember that with an HSA, the tax deduction comes when you put money in (contribute), not when you take money out. You have already gotten your tax deduction when you put money in. If you end up taking out $600 that you then have to put back in later as a mistaken distribution, it doesn't affect your taxes in any way. This is true even if it falls over different years: If you take out $600 to pay for a medical expense in 2021, and then get reimbursed by insurance in 2022 and have to send the money back to the HSA, it does not affect the amount of your tax bill.
Alternatively, if you prefer, you could pay the $600 with your own non-HSA money (from your checking account or credit card). Then, after insurance reimburses you for most of it, you could seek reimbursement from the HSA for the difference. There is no time limit for requesting reimbursement of a medical expense from the HSA.