No, the software is working correctly, even if the intermediate steps are not obvious. The way it works is this:
HSA contributions made by your employer or by you by means of payroll deduction are reported in box 12 of your W-2 with a code of W.
The amount is also removed from Wages in boxes 1, 3, and 5 before the W-2 is printed. This is the benefit of contributions made through your employer - the amount is never in your income in the first place, and your don't pay federal income taxes, Social Security taxes, nor Medicare taxes on this amount.
However, the IRS considers HSA contributions to be taxable until you show that you have proper HDHP coverage.
So, at the time you enter the W-2, the code W amount is added back to income (line 8 on Schedule 1 (1040) if you want to go look), until such time as you finish the HSA interview and show that you had HDHP coverage.
At that point, then, the code W amount is removed from line 8 of Schedule 1 (1040), and your taxes go back down.
This is one of many reasons why it is not a good idea to watch the Refund Meter too closely - there are things happening in the background asynchronously to what you are entering, but it will all come out good in the end.
P.S. The reason that TurboTax 2021 appears to work is because you had already entered the HDHP information when you did the return last year, so TurboTax didn't add the income back. If you had started with a blank return for 2021 (which you can't for 2021 because the current Online system is for tax year 2022 only), you would have seen the 2021 software do the same thing.
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