Glad you found my post interesting. I found yours interesting as well, as I thought I was the only one who made any argument along those lines.

There's no explicit step where you say the speed of sound is infinite. That's just the net effect of how you model the pressure field. In reality, the pressure comes from thermodynamics at some level. In the low-Mach/incompressible model, the pressure only exists to enforce mass conservation, and in some sense is "junk" (though still compares favorably against exact solutions). Basically, you do some math to decouple the thermodynamic and "fluctuating" pressure (this is really the only change; the remainder are implications of the change). You end up with a Poisson equation for ("fluctuating") pressure, and this equation lacks the ability to take into account finite pressure/acoustic wave speeds. The wave speed is effectively infinite.

To be honest, I need to read papers like this to gain a fuller appreciation of all the implications of this approximation. But what I describe is accurate if lacking in some of the details.

In some ways, this does make things more complicated (pressure boundary conditions being one area). But in terms of speed, it's a huge benefit.

Here's another example from my field: thermal radiation modeling. If you use ray tracing (like 3D rendering) then it's often practical to assume that the speed of light is infinite, because it basically is relative to the other processes you are looking at. The "speed" of heat conduction, for example, is much slower. If you used a finite wave speed for the rays then things would be much slower.