For 2022-2024 you are correct, you must amend and pay the 6% penalty each year.
For 2025, the HSA bank is correct, it is too late to use the special procedure to remove the excess contributions.
There are two ways to remove the excess contributions.
1. Make a withdrawal of the excess amount (ignoring the earnings, just the contribution amount). Do not use it for medical expenses. On your 2025 tax return, you will be charged income tax plus a 20% penalty. That will remove the excess from the account for the future.
2. If you are eligible for HSA contributions in the future, you can apply the excess to your current year eligibility. For example, if you enroll in a qualifying HDHP for 2026, you would not make new contributions, but instead the 2026 limit (probably will be around $4400 for single and $8800 for family, although the amounts have not been announced) would be applied to the 2022 and 2023 contributions, using some of them up. You would then pay a reduced 6% penalty on the remaining excess, which you could use up in the same way in 2027.
Because of the hefty tax and penalties on a regular withdrawal (method 1), if you think you will be enrolled in qualifying insurance in the next 4-5 years, it will probably be less costly to pay the ongoing 6% penalty and eventually use up the excess when you are eligible. If you don't plan on enrolling in an HDHP in the next 5 years, you should probably remove the excess, as painful as that would be.
