Well, yes except there are no routine emigration checks when leaving the US or Ireland, so when leaving Ireland you need to show your US passport to the airline otherwise they will not let you board a flight to the US. Not so much an issue in the other direction for the moment. Of course if US CBP decides to do a random passport check on leaving the US you should show them your US passport.
Again true except with the current case in point, a US 'visa waiver' does not cost nothing.
Well I am not sure what you were trying to communicate but whatever it was certainly isn't clear. If you do not use the same passport when leaving a country (any country) as you used when you entered that country, then it is possible to end up being flagged as having 'overstayed'. That will then mean having a problem if you ever decide to return to that country. Let me ask you D!, do you have 2 passports and have you travelled much using both in one trip? In regards to this specific post, the rule that applies is the law that says a US citizen MUST leave and enter the USA on a US passport.
Regarding a US visa waiver, what has a US visa waiver got to do with what I wrote? Some countries require that you pay for a visa or for a visa waiver, like the USA. Some countries make you pay for a visa but not for a visa waiver. Therefore, there are cases where entering on a passport that gets you a free visa waiver entry is advantageous. Do you not understand that or do you think talking about a country which requires you to pay for a visa waiver means there is never an instance where you can save a few dollars entering a 'free' visa waiver country? In regards to this specific post, there is no US Visa Waiver involved as the traveller is a US Citizen.
By the way, 'does not cost nothing', is a double negative and makes no sense whatsoever. Proper English would be 'doesn't (or does not) cost anything', which is not what you actually wanted to say. You meant, 'does have a cost' or 'is not free'.