Why the lack of faster piston-powered planes?
The power requirement of an airplane grows with the cube of speed. When you fly fast with an airplane which needs to comply with a set minimum speed imposed by regulations, your drag coefficient is nearly constant, so flying faster does two things to a propeller aircraft:


Drag goes up with dynamic pressure, which is proportional to speed squared
Thrust goes down with the inverse of speed.


Power is thrust times speed, so if the the 350 hp get you to 210 knots, doubling the installed power can only bring you to 264 knots. A turbine can still make limited use of the increased dynamic pressure by ram recovery, so here the thrust decrease over speed is not quite as bad as for piston engines.

Expressed in an equation, the power $P$ demand for a given speed $v$ is:
$$P = \frac{D\cdot v}{\eta_{Prop}} = \frac{\rho\cdot v^2\cdot S_{ref}\cdot c_D\cdot v}{2\cdot \eta_{Prop}} \rightarrow v = \sqrt[\Large{3}{}]{\frac{2\cdot P\cdot\eta_{Prop}}{\rho\cdot S_{ref}\cdot c_D}}$$

If you want to fly fast with better efficiency, you need to increase the minimum speed - note that the landing speed of many fast piston aircraft was around 100 mph. Now you need a long runway, and for operation in bad weather or at night infrastructure for instrumented approaches, and you will end up at airports where turbine fuel is easy to get and cheap, but piston fuel will be hard to find and expensive.

As with all expensive machinery, you need to use it often to justify the expense. Now your fuel cost will factor in, and will make a piston engine rather unattractive.

Fast piston aircraft were only built while turbines were not yet available. As soon as turbines emerged, all fast piston designs were obsolete. And aerodynamics in the early 1940s were already very advanced; engineers back then built airplanes which would be impossible to design with the engineers of today. Back then, they got fully manually controlled airplanes into the air when today everyone would refuse to do it without hydraulic boosters at only half the flight speed. They could design cooling ducts which actually increased thrust, an art which is (almost) lost today.