How/Why Do Bode Plots Work?
This is how I see the bigger picture...

The transfer function and bode plot are intimately related. More specifically the pole zero diagram and bode plot are related. See this example of a 2nd order low pass filter: -



Top three diagrams show the amplitude part of the bode plot. Bottom left shows how this fits into the 3 dimensional picture and bottom right is the plan view of the 3D picture (pole zero diagram).

So when you replace s with jw you are just focusing on one aspect of the 3D big picture.