You're correct, a new patient visit 99203 is for the evaluation and management of a problem. A chief complaint or presenting problem is required, and that is true even if it is being billed based on time. The provider may be correct that if the patient did not come in for a physical, or documentation doesn't support a physical exam, then charging for a physical would also not be correct.
So I guess the question is, if the patient does not have a problem and did not request or have a physical, why are they being seen and what is being done? Is this a 'meet and greet' or visit to establish care for a healthy patient? That is not something that is covered by insurance as a medical necessity or benefit under most patient plans and should either be a no-charge or a self-pay service. Or is the provider taking over medical management of the patient but there is no current symptom? If that's the case, then the provider should be documenting what chronic conditions the patient has that will require management, and that can qualify as your problem/chief complaint for a new patient visit.