Keeping the gas going in the "right direction" is everything to do with propulsion.
The starting cycle is a simple way of understanding the need for compression. Without compression, there is no work. How can compression become irrelevant to the process at any time?
Starting is compression, it contains gases in a dynamic flow by directing its passage from a large to small annulus. Without the pipe, there is no power, and without the fan, there is only multidirectional expansion.
Water injection, Open Iris Afterburner, Hypergolic starting, Start carts, Compression is our friend. The difference between early centrifugal compressors and axial flow instructs as to the OP post. PUSH, PACK, POP, off you go.