It seems to me that this is all getting overcomplicated - AND back-to-front, as it were.
The simplest answer to "Why a compressor?" is - to provide "artificial airspeed" through the engine so that it will run even when sitting still. Actually, of course, it is REAL airspeed - but localized to within the confines of the engine.
Without that flow - you get, as previously mentioned - a bonfire. Google "turbine hot start."
Pure internal-combustion jets (e.g. ramjet) require substantial forward speed (= airflow through the engine) to function at all. At least 100 mph (160 kph) for minimal function. "Best" operational speed is at Mach 0.5 or higher.
Very hard to taxi (or pull up to the gate) while maintaining Mach .5
So - there needs to be a way to keep the engine up to minimal self-sustaining spin, even with little or no external inflow of air from the speed of the aircraft/vehicle itself. Thus the compressor.
A ramjet works with neither compressor NOR power turbine! The main function of the power section in a turbojet, at least at lower aircraft speeds, is - to drive the compressor to sustain ignition! Ideally subtracting as little power from the exhaust thrust as possible.
In a turboprop/turboshaft engine, that is the main function of the power turbine - period. A turbine helicopter can hover while the compressor keeps the airflow/speed through the engine up in the mach numbers.
If an aircraft reaches a high enough speed, in theory one could fold the compressor blades out of the way, and just let the ram air do its thing unimpeded. Hard to build a hinge strong enough to take the stresses, though, and in fact, development went the other direction (fanjets and turboprops) where the compressor (or at least a part of it) becomes the primary source of thrust.
Now, there is lots of room for theory and engineering to make the compressor (and the turbine driving it) work as effectively and efficiently as possible, which is the reason for all the theory and measurements and charts.
But that answers not "Why a compressor" - but "HOW a compressor?"
@ FS (below) - yes, OK, I was referring to compressor as the physical bit of machinery. A ramjet does need compression - from ram airspeed, augmented by venturi compression (usually). Just not the spinning fan.