Unfortunately, either what you read was wrong or you misunderstood it.
What you may be thinking of is that if you take a distribution for nonqualified medical expenses. you will (1) have to report that amount as income, (2) pay a 20% penalty on top of that, but an amount of the excess carried over from a previous year equal to the nonqualified distribution will be forgiven, well, discharged.
Note that you can withdraw excess contributions up until the due date of the return, so 2020 excess contributions can be withdrawn until May 17th (October 15th if you file an extension).
Once that date has passed, you can no longer withdraw this excess. So for an excess from 2019 or before, you must discharge it either by applying it against a future HSA contribution limit if you are eligible to contribute or by taking a distribution for non-qualified medical expenses.
Please show me the link to what you are reading, and I'll explain it to you.
P.S. "Otherwise, the same excess contribution would be taxable every single year. " - in fact, this is the way it works unless you either discharge the excess in a future year or make a nonqualified distribution...or your HSA runs out of money (then the excess is still carried over but the penalty is reduced to zero).
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