My father flew at the cusp between jets and propeller aircraft. He said exactly what you said. The early jets were dogs until they spooled up but obviously at altitude they were far superior to the prop planes.
His favorite plane was the F8F bearcat which was a monster and at low altitudes (below 20,000) could eat a P-51 up. He said they used to jump the air force guys and eat them up. Once the P-51 got above 20,000 or so the tables turned.
How about that? So did my Dad! He was an aviation machinest's mate in the Navy from WW II through the late 50s. His boast was "I can take apart and reassemble a Wright Cyclone radial engine blind-folded!" He went through the transition from piston to turbine era.