When the HOA (or the county clerk on their behalf) puts the property up for foreclosure auction, they’re basically putting their judgment up for bid. They can either set the opening bid at whatever they feel makes sense, or they can come in and actively bid during the sale up to the amount they’re owed.

Once the bidding is over, the winning bid is considered payment toward their judgment. In other words, they’re accepting that final bid as satisfaction “in lieu of” the full amount they were awarded in court. If the bid comes in short of the judgment, the HOA eats that loss, you as the new owner don’t get stuck with the unpaid balance.
If they want to try to recover the difference, their recourse is to go after the former owner personally for a deficiency judgment, or, more commonly, to spread the shortfall across the rest of the community in the form of increased dues or another assessment. That’s how the HOA makes themselves whole, not by coming after you.

So as the investor buying at auction, your responsibility starts fresh the day you take title, you’re only on the hook for future dues, not the past $125K. The key is making sure there’s no senior mortgage or other liens still hanging around that could trump the HOA’s foreclosure.