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There are all sorts of bridge shape: the straight approx. 1" wide type with pyramid ends, the current Martin design, and any number of one-of-a-kind designs with fanciful end treatments.

How much does any of it matter? Why choose one design over another?

I'm using a pyramid bridge for my guitar build, but aside from aesthetics I have no reason for that choice.

what matters most is
1 is the glue footprint sufficient for the load
2 is the design too large
3 is the design too small

so you have an inference from the bridge some helpful some hurtful. It is about mass and physics. So yes the bridge can influence what the top will do but is isn't that simple. You have to take into account the plate braces and bridge. This is a complex area.

I prefer smaller bridges like the 1by 6 rectangular or pyramid for smaller bodies and the belly style on dreds. On lighter braced guitars like 1/4 in I will use a belly on small bodies after all the bridge is the only brace outside the box.

The width of the bridge and the underlying plate will affect how easily the bridge will rotate under load. The bridge and bridge plate are two parts of an assembly. One can decrease the width of one part and maintain the same functional aspects by increasing the other part.

In addition to width of the bridge, length is also a major concern and at least as important. How the wings connect/overlap the braces underneath has a direct influence on how the system as a whole accepts the load applied by the strings and eventually the variances in load caused by string vibration to produce sound. If the bridge is too narrow and falls short of the X braces the bridge will likely fail because the load is not being transferred to the actual structural part of the top. If the bridge is too wide and protrudes to far beyond the X it may start to actually choke the vibrations of the top by creating a larger rigid surface area that cannot vibrate freely.

Another consideration needs to be the flexibility of the wings that are over the braces. They need to be flexible enough to transfer vibration to the underlying structure yet stiff enough not to fail under load. A belly bridge has thinner flexible wings while a pyramid bridge has a cut between the center and each pyramid to provide a type of hinge and give flexibility in that way. Each doing basically the same job in a different ways.

Then there are a whole host of other things like mass, height, plate material & thickness etc. that also should be considered. The bridge is where a lot of it all happens. It is where the energy of the strings is transferred to the actual sound producing parts of the guitar. In simplest terms it is a lever and as such there are many designs that will work. But careful thought and planning in this one area can make all the difference between an OK sounding guitar and an exceptional one.

- Footprint matters as it affects gluing area. The depth front-to-back comes into play as well in the glue resisting the pull of the strings.

- A larger footprint affects stiffness in the top. Wide bridge affects cross-grain stiffness more and and a bridge deeper front-to-back affects long grain stiffness more

- Bridge weight matters and affects tone. So the shape of the bridge can add or minimize weight affecting tone. Within reason and assuming the bracing system is setup with this in mind, most folks think a lighter bridge is better as it has less impedence so it doesn't impede sound as much as a heavier bridge. May have a better "attack" as well (subjective point).

- The bridge size in reaalting to the bracing underneath matters as it's a system, not isolated functions. For example, how a bridge overlaps the X-brace and the stiffness of the area that overlaps affects the overall stiffness of the top.

So I say it does matter.