Border guards check your passport stamps to see any previous record of overstay in their country when deciding to permit or deny you entry.
If so, then what do they do when you present a newly issued passport that has zero passport stamps? All sins forgiven (hidden)?
Almost every country has computerized system for border controls now. So, whenever they scan the passport front page, all the details are immediately available on immigration officer's computer screen. Based on that they will ask questions and decide if you can be allowed entry.
It is trivial to match you with a previous passport by matching your name, date of birth and photo. This will be the case even if you acquire a different citizenship. Some countries list the older passport numbers in back page, to make this easier. They may even have your fingerprint.
So, in any case, only the passport is new. You are still you and your sins are still in their system. If you lie, they have mechanism to find it and the outcome will be much worse than telling the truth the first time. You may sometimes get away with this, but risks really outweigh any benefits you may obtain.
Notable points
then what do they do when you present a newly issued passport that has zero passport stamps?
For the Schengen Area: nothing, since after 90 days outside, the border control is not interested in the previous periods you stayed in the Schengen Area.
If you are leaving the Schengen Area, without an entry stamp, you are required to prove when you entered (otherwise an overstay will be assumed).
Once the Entry/Exit System (EES) is introduced (possibly 2024), passport stamps will no longer be needed since the travel history will be stored electronically.
The original passport conventions never foresaw that the 'Visa' pages would be used as a travel record.
Until after the 1990's, it was unheard of for a country to ask for a previous passport for a visa application.
Some countries even didn't allow people to keep a previous passport, so supplying one would not have been possible.
Things evolve with time.
Soon the present day passport books will be replaced with some form of passport card based on ICAO Document 9303 ID-1/TD1, which most European ID cards allready support (National identity cards in the European Economic Area - Wikipedia).
The Babylon 5 Identicard will then become reality.
If so, then what do they do when you present a newly issued passport that has zero passport stamps? All sins forgiven (hidden)?
You have to make a distinction between your legal position and how enforcement works in practice. Getting a new passport only changes how likely border guards are to realize that you have overstayed but it doesn't mean you have a right to stay longer.
Border guards can still look up information about your comings and goings in any computer system they might have. As explained in the answers to your other questions, passports are not used for that and this scenario provides a very good reason why it would be a bad idea: You don't want to make the retention of this information dependent on random actions by foreign countries like a renewing a passport.
If there is a formal ban, you're very likely to be found out based on your name and date of birth (that's kind of the point of a ban and one difference with simply being denied entry).
Outside of these situations, it's entirely possible that border guards fail to notice, say, an earlier stay or even a violation or overstay after you changed your passport. It doesn't mean that you have the right to stay longer than you would with your old passport but simply that you're getting away with something that would otherwise have resulted in being denied entry. Think of it like speeding in a place where there is no speed camera or parking illegally.
All this is especially relevant in the Schengen area. Since the borders between member countries are open and there is limited information shared between countries, the Schengen Borders Code traditionally relied primarily on passport stamps to enforce maximum stay rules. A border guard can easily be unaware of an earlier stay in another country in the area. This has been bothering people for a long time and that's the reason why the EU is rolling out the Entry-Exit System.
