It works exactly like your "linear" example of gas turbine theory except that, by putting some processes in parallel and reversing the direction of flow, it is shorter.
Imagine, if you will, the digestive tract of a 3-metre python. (I know, but its a McGuffin, stay with me). Lay the python out in a straight line and feed it a rat; it eats, digests, poops. Now let the python adopt a coiled, snakey, posture and feed it another rat. It eats, digests and poops, but its now a one-metre python and will fit more conveniently inside the cowling of a King Air.
Getting back to the PT6, all the components and processes of the linear gas turbine are present. Having passed through a centrifugal compressor the annular gas flow has a conveniently large diameter, large enough to accommodate a turbine within it. Yes, the compressor discharge air flow reverses, but retains its pressure and velocity as it enters the combustion liner through the myriad of cooling holes you can see in Post #3. Combustion gas then reverses flow again, but the annular flow now has a smaller diameter so that it, and the turbines, fit within the combustion section. Same processes, same cooling theory.
Edit; I've re-read the OP and looked again at the diagram; its a gross simplification. The light-blue compressor discharge air does not just flow into the side of the combustion liner as illustrated. It discharges into the complete void surrounding the liner and enters from all directions; inwards, outwards, backwards.