Earlier quoted context omitted.
The semantic error you're making is redefining the general term compression as the more specific "isentropic compression". The process by which the compression takes place is not prescribed just by the term "compression" as you did originally. Just saying "compression" specifies nothing about the process by which it happens. Compression can take place (theoretically) through an isentropic process, or a non-isentropic…
Consider the following extension of this thermodynamic process. The gas is shocked, and then we adiabatically expand it back to its original density. Because entropy was added, the gas is now hotter than initially. But it is not compressed. The heating wasn't due to compression, it was due to dissipation at the shock. That dissipative process is separate from compression, and you are confusing the two. The incoherenc…
The position is far from incoherent. Temperature rise during isentropic compression is well understood, as is the temperature rise and total pressure loss through a shock.