Yes, 2 wheel drive (either front or rear) is all you need for everything except winter driving.
In snow you are generally much better off with front-wheel drive (engine weight is over pulling wheels). This category includes common rental vehicles like a Toyota Camry or Corolla, and Honda Accord or Civic.
A 4X4 (Jeep etc.) is not necessary at any time of the year, but is helpful in snow.
Nothing on Route 66 involves off-roading. Even in areas such as the Cuervo Cutoff and La Bajada in New Mexico, you can park nearby and explore on foot.
(This is not a timely reply to tezza, but it's a question that keeps coming up.)