I know I'm being a bit simplistic, but if you use energy to compress a gas, you regain energy as it expands. So all you need is a timed breather to cope with the excess gas from piston ring blowby and the majority of air in the crankcases is just squeezing and expanding and staying almost but not quite energy neutral.
If you are pumping it continually, might you not be expending more energy?
In the mythical engine used by schools to teach the relationship between Pressure , Volume & Heat you would be almost correct if you ignore the heat radiated out of the engine.
In the real world not quite so.
Unless you are using total seal rings in both grooves you get leakage between the rings & the bore / piston.
So you always get gas bypassing the piston. More on the compression stroke than the other 3 .
Next pumping a fluid requires a lot less energy than compressing one.
But don't believe me .
Go to the blog and see the results form the research engines.
There are other benefits from through breathing but they are a bit on the esoteric side.
The basic problem was the original breathing system was designed to SUCK AIR INTO the engine past all of the oil slingers because they did not use seals on the crankshaft.
This gets overlooked by most people because they do not look back far enough , just at the problem as seen today.
Because BSA's were designed in the slide rule days the methods were different and usually it was a case of using what they knew from experience worked on the previous models.
When you have a bad case of Beeseritus so play with & learn about everything from the year dot, it becomes apparent.
You see things like bore & stroke ratios that carried on from model to model, because they knew that this ratio worked and would not crack frames or set up sympathetic vibrations that sent the bike sideways at specific revolutions.
In the computer in your pocket days where almost any idiot like me can design an engine then run it without touching a piece of metal, it is difficult to appreciate the problems back in the 40's & 50's when the A7/10 was deigned and every thing had to be thought out then made in metal to test your idea.
For BSA engine breathing was not a problem they just did what they did with the previous model, till they needed to get better performance from the B44 engine in order to hold on to the only area of motor sport where BSA was still a power thus got the results to promote sales. So the B50 got the closest thing to an adequate breather that BSA ever fitted to an engine.
The B50 engine was pumped up to the highest compression ratio that would stay together, but they needed more Hp and they got that little bit extra by the 1/2" breather.
Ten years latter Yamaha brought out the SR 500 which also used the massive 1/2 " breather to squeeze that bit extra out of the engine.
Had either of them put a timed or controlled inlet on the rocker cover they would have seen another Hp or 2 from that engine .
The easiest way to increase the Hp on an engine was to bump up the compression.
Following that was gas flow within the head so that is where all of the research went.
By the 80's EPA regulations world wide was pushing towards controlling oily gas emissions by forcing the crankcase gasses to be passed through the carb & burned so research on through breathing became a non event.