A lot of these are partially correct, but not entirely. Especially in the case of dual citizens travelling between their two countries of citizenship you'd encounter a very un-fun experience.

For a start, as you recall the gate agent will scan your passport through the reader before letting you on the plane - it will appear with either a green light which you no doubt recognise and an "OK TO BOARD" response, or a red light and "DO NOT BOARD". This is checked - in real time - with the border agency of the country you are travelling to. If you do not have an API record for the plane, this will usually result in "DO NOT BOARD" and the agent must request a "Governmental Override" from the border agency.

Typically for a dual citizen, they in many cases actually present both passports at each end of travel, as they must leave each country on the passport they entered on, and cannot travel on each other's passport to the other country, as they cannot be granted a visa (and therefore will be denied boarding by the API check). Another option is to present the exit immigration control with the opposite passport to the one that is presented to the airline (and included in API). This carries risks where the API is used by the immigration agency to determine your exit (ahem, USA) - there is actually a working group proposing multiple passports in an API record to solve just such a problem.

Incidentally, an ESTA/ETIAS/ETA isn't really a yes/no result as to whether travel is allowed, it's merely permission to approach immigration control and request entry into the country. You can still be denied, and returned to your origin country at your own expense.