Yes, you definitely need to enter the US on a valid US passport if you have one - and airlines will often refuse travel to you if you turn up with the wrong passport eg a UK and not a US passport at LHR attempting to travel to NYC.
From a career at LHR long past, it was amazing the number of people who were unaware of the rules, despite their children being dual UK/US nationals. I've had many a debate with an irate mother or father, whose children had out of date (and hence invalid) US passports and wanted to go to the US on their UK documents. Yes, technically, we could have let you on the 'plane - but you'd have been turned back by the US authorities and my airline would have been billed with repatriating you. Hence, we didn't aid and abet breaking the rules!
I'm a bit surprised at the blasé attitude lots of people have to this kind of thing in general, especially with the relatively new tax rules that came in for those born in the US recently. My next baby will be born in the US and we have done lots of research into what this means - so baby will need to be issued with a US passport first, then a UK one either through the embassy or on our next UK visit. DH, DD and I will have UK passports/visas to travel into the US on, new baby will have a US passport. Similarly, DH and new baby will have SS numbers, DH through work and new baby through being seen as a US citizen. DD (UK born) and I are not entitled to SS numbers, but rather ITINs in order for our tax return to be processed.
Once new baby reaches 18, he or she can decide whether or not to retain the US nationality. Yes, in some ways this will open doors, but in others, eg if he or she were to work in a country where earnings were significantly higher than the US, he or she would then end up owing tax to the US government. This doesn't sit easy with me, all due to incidence of birth, but them's the rules at present, whether we like them or not.