For years I used to design 2 layer boards with whole bottom destined to one continuous GND zone. So all signal and power lines were at top. From time to time I had to use some 0Rs.
Here is the example how it looks practically (all vias are GND):
Now when I started to design 4 layer boards I have In1 and In2 GND, signals at top and VCC at bottom (I have only VCC and GND vias). But my designs (as you can see in the above example) are simpler to be routed than yours. But even with your circuit there probably will be no problem to route signals and power having 2 layers for it.
I have a strong preference for a 4 layer board with 2 ground planes, over a 4 layer board with a power and a ground plane. The strongest reason is that both ground planes can be stitched together with via’s, so interruptions in one ground plane can be repaired by the other ground plane, so the loop area of the return current is minimized.
To answer the question, I would try to put the power traces at the top or bottom signal layers, and then put short power traces in one of the middle layers then it’s not possible to route it all on the signal layers.
Every time a trace is placed in the ground plane layer, it must be stiched with via’s, so he return current has as little loop area as possible.