The German engines used a different approach to getting power than the Allison or Merlin. Airplane designers do not care what the HP per cu in or cu liter is. They do care what the horsepower per pound of engine weight is. You can build a large displacement slow turning engine for the same weight as a small displacement high rpm engine. Picking either approach is not "thinking out side the box" as this choice had been faced by engine designers for may years. The availability of materials such as good bearings at a given time can influence the designer one way or another. As could previous experience (in the late 20s the internal combustion engine had been around for just 30 years) or a designers own ideas/ prejudices, as in air cooled or liquid cooled. designers or even companies often picked one or the other and stuck with it.
Once a designer/team starts down a certain path certain developments follow. Some of these developments over improvements in one area or aspect (or more) of performance but have limits or drawbacks in others. The Germans were very interested in fuel efficiency, as well they should be given their chronic shortage of fuel. High compression helps here, but high compression limits the amount of boost that can be used with a given grade of fuel. The Merlin used a 6:1 compression ratio while all but a few Allison's used a 6.68:1 if memory serves. It was estimated that the Allison got 5-10% better fuel economy than the Merlin but couldn't use as a high a boost pressure. At the end of the war (and post war--the P-82 twin Mustangs) some Allisons were fitted with 6:1 pistons to allow for higher boost and power levels.
Airplane engines are very lightly built. The stresses on the Crankshaft and bearings go up with the square of the engine speed (rpm) increasing the max RPM of an aircraft engine (and keeping anywhere near the same engine life between overhauls) by even a few hundred rpm usually called for a new crankshaft and a strengthened crankcase. Cylinder size also plays a part in rpm limits. The gasoline mixture in the cylinder burns at a pretty constant rate, that is if ignited on one side of the cylinder the flame front travels across the cylinder at pretty much the same speed in all engines. Too wide a cylinder at high rpm means that the burn isn't completed when the piston reaches bottom dead center or the exhaust valve opens, which ever is first. Big heavy pistons also don't reverse direction very well. Too long a stroke for a given bore size goes the other way, combustion is finished (for the most part) before the piston gets near bottom. While the gases are still expanding some what peak pressures are long gone. It might be good for fuel economy but not for high power output. These are generalizations and don't really apply to the the engines we are discussing because their designers knew about this to begin with. It does help to explain why the Russian AM 35-42 series was the largest V-12 used in WW II though and it used the same bore and stroke as the BMW V-12s used in many of the Luftwaffe's early airplanes. See:
BMW VI - Wikipedia, the free encyclopedia
Notice the weight. Large, light,slow turning engines were not new, they were not "outside the box". In fact they actually represented the normal in aircraft engine design when you consider the Hispano engines and the Italian V-12s and W-18s.
In the 1930s with 80 and then 87 octane fuel every bodies superchargers were pretty much the same. The fuel just wasn't going to support much more than 6:1 to 7:1 compression ratios and 4-6lbs of boost. The German move to fuel injectors may have been an attempt to push that boundary at bit. It also offered less chances of icing in the intake, freedom from backfires and a few other advantages. When better fuel came along and superchargers could offer higher boost (Early Merlins and Allisons got a good part of their power from RPM not boost) the British and Americans were able to use the evaporation of the fuel in the supercharger as a low level charge coolant. I would guess that this was more by chance than a long term plan. Once noticed however it was a strike against going to the German style injectors if other problems could be solved.
Since it can take 4-6 years (or longer) from start of pencil on paper to squadron service of an aircraft engine decisions made in the mid 30s affected engine choices and design till the end of the war if not beyond.
I am not trying to say one side was better than the other or smarter.
Choices were made early on that lead to other choices later or that limited options in one way or another later on. There was little that was "NEW" in engines as far as ideas went. Turbo chargers had been used (experimented with) since WW I in a several countries. Diesel aircraft engines had been worked on by the Americans, British, French and Russians as well as the Germans (maybe the Italians too?) Fuel injection systems had been worked on by a number of countries/companies. The USAAC wanted Allison to work on fuel injection in the mid 30's, Allison turned down the request because they didn't have engineers to take on the project. Bristol in England had several fuel injected prototypes in the early 30s. Water injection had been tested in the Mid 30s. exotic fuel blends had been a staple of both auto racing and speed record aircraft flights for years.
The Germans were not "thinking outside the box" any more than anybody else was to end up where they did. They may have devoted more resources to a particular area than other people did while other countries/companies devoted their resources to a different aspect/area of engine development.
As an aside the P-38 troubles with fuel distribution were known, predicted and a solution was in the works even as it happened. The Specifications for 100/130 fuel was changed allowing for a higher amount of heavier compounds which would separate out at low temperatures. This could potentially affect ALL Allison engines, not just P-38s although due to the altitudes at which they flew they were the most affected. A new intake manifold was designed to solve the problem and was being fitted to engines at the factory in Nov/Dec of 1943. Hundreds were shipped overseas for refitting to exiting engines. Most of the problems occurred in those few months while the change over was taking place. Work had started on the Manifold back in the summer of 1943. This is part of the problem the US and to some extent the British faced. It could take weeks if not 2-3 months to get a factory modification to units in the field.