The way you have framed this question, you make the "black and white" answer seem to be: come clean and announce to the world publicly that you committed some kind of misconduct with the papers you wrote and issue a correction.
However there are several factors here that I am reading between the lines that make me doubt that the "black and white" answer really applies here.
Guilt -- especially guilt over something that has been festering -- can skew your perspective on how important something is. It's not clear to me, in objective terms, how serious the ommission is, but my suspicion is that it is fairly minor. To me, "major" would mean that the main result of your paper is not reproducible even in principle; "minor" would mean that you did not provide enough information to fully reproduce the results but that the main conclusions are correct and reproducible (perhaps with some additional input that can't be reconstructed from the text of the paper).
Our own faults are very clear to us, while the faults of others are often less obvious. Are you sure that your assessment of your work is fair, in the sense that other papers in the literature do not contain similar omissions/simplifications? This is particularly the case since according to your description, your standards are higher than many of your peers.
Science can be brutal, and we are hard on ourselves. People make mistakes, and deserve the room to learn from those mistakes. Be kind to yourself.
Furthermore, it's important to be accurate in your assessment of this, because "going too far" in the direction of "apparent good" can lead to damage to your reputation -- if you tell people you are guilty of something, people will tend to take you at your word. Admitting to falsifying results in a paper, even approximately falsifying them, is a pretty major offense in the scientific world.
I think you need to evaluate as honestly as possible, without guilt, whether the papers currently in the literature actively causing harm. Are people citing these papers? Are they building experiments based on the parts you know are wrong? And... realistically, do you believe your omissions are significantly larger than others in the literature in your field? If not, then let it go. I promise you, you are not the only one who has done something like this, and from your description I seriously doubt that what you did is the worst thing that exists in the scientific literature.
If there are really wrong results in those papers and those wrong results are being used as a basis for follow-up studies today, then that does make the situation more complicated. But I would still proceed cautiously. I would publish an erratum or some updated paper addressing the issue, but I would strongly recommend to keep the focus on the science, and not admit fault or falsification of the results. Just state that you became aware of errors or omissions and you are submitting a clarification.
The problem you have is that by raising the issue now, particularly if you frame the issue as something serious like "falsification", years and years after the paper was published, is that (a) people will wonder why you are brining it up (they may question your motives), and (b) you bring attention to the issue. If harm is not actively being caused, and your guilt is causing you to overestimate how significant the actual offense was, then there is a serious risk that the net result will be self-sabatoge without helping yourself, anyone in community, or, I would argue, even "the cause of justice" in an abstract sense.
You seem to be a very serious and dedicated scientist. Focus on doing good science and bringing positive and exciting new results into the world. Don't let this consume you.