The reason that I say this is because there was never a time when an American president clearly stated what the US was going to try to accomplish in Vietnam and how far it would go to achieve its goals.
The US involvement started small -- with aid to the French. Clearly, US policy was to try to stop Vietnam from falling out of French hands and into those of Ho Chi Minh and the Viet Minh. But how far would the US go to achieve this goal? This was not spelled out. Pres. Eisenhower, for example, did not say in 1954 "we are going to support South Vietnam all the way up to the point of having 600,000 troops in the country." Instead, things just sort of went along.
For example, it was not Pres. Eisenhower who said that we would support Ngo Dinh Diem and who said exactly how we would do that. Instead, it was the CIA who worked to support Diem. This decision later backfired on us as Diem became more and more unpopular and we were already committed to backing him.
American involvement in Vietnam grew gradually with decisions like that. It did not grow according to some grand plan set out by a President.
Could Presidential involvement have caused a better result? That is by no means clear. But direct presidential involvement might have resulted in clearer policy with definite goals instead of the sort of "mission creep" that we ended up with.