This is why the IRS sending you a pre-filled return would help with such a situation. Right now the employer can make mistakes and you don't even know.

Similar example from Germany:

In one year I had some "extra income" besides my job and had to file a return and pay some extra money.

The next year they asked me what my "extra income" was for that year and when the letter got lost by me they just assumed the same amount as the prior year and sent me the bill for that.

After the initial shock and beating myself up for that mistake, I could simply file a correction and got it fixed with no problem.

Is there a website you could just have updated, or does that require paper forms? Randomly curious.

I'm not sure and it was a few years ago.

I faintly recall being able to do it online but there is a fair chance that I missremember.

In which case you'd have the possibility to contest and file something more specific to your situation.

Your suggestion sounds like the current system: I have to calculate the correct taxes and file them based on the truth.

Well yes, if you think the numbers the IRS calculated aren't quite right you'd still be free to file your own return.