You should be fine renting a car with a U.S. license. You might want to check with the rental company about their minimum age requirement though -- often 21, sometimes 25.
I'm afraid that Britain's public transport network is nowhere near as comprehensive at it used to be in the past, and there are many little places where a car is really only practical way to get there.
On a historical note, many miles of small railway branch lines were closed down and torn up in the 1960s. In some rural areas you can still see where the line used to run, and in fact there are sometimes designated footpaths which follow the old right-of-way and are popular with hikers.
If you see a new-ish looking road in a town with a name such as Beechings Way, have a look around the neighborhood and you're sure to find some evidence of an old station or rail line. The name is a reference to Dr. Beeching, the government minister who was largely responsible for the closure of many branch lines.
The buses have fallen into decline rather more recently. It used to be the case that companies had to run on non-profitable routes as a public service, effectively subsidizing these services with their busy routes. Since deregulation of buses during the 1980s/90s, services have been decimated in some areas.
The joke where I live is "A bus? Oh yeah, I remember, those big red things........."
