Yes, as a US citizen you have to enter and leave the USA on your US passport. I was also advised by UK Border Force that I should enter the UK using my UK passport as that prevents any issues with potential overstays if I enter on my US passport.

Regarding the requirements for an ETA to visit the UK, I applied for an ETA using my US passport / citizenship and under the "Other Nationalities" selected "United Kingdom" and then "British Citizen" and the application was approved the following day. This means I can complete any API data using my US passport and (should) prevent any OLCI challenges, but I can then still use my UK passport when I reach the UK border.