You're right, the speed is not the same. Extending your thought experiment with measurement equipment may help to understand in more detail.
Consider a sound receiver which activates a light that the sender can see. The local propagation speed of light in air is near $c$ (and $>> V$) so we can consider the light speed signal instantaneous for the purposes of measuring the speed of sound between the two points on the platform. This avoids a need to invoke outside observers, letting these observers directly measure the speed of sound.
(The signal could be a momentary chirp, with the receiver lighting up an LED after detecting a few oscillations at that frequency. Or it could be a modulation (such as AM or FM) of a carrier audio frequency with the receiver demodulating, and encoding that info onto an LED in some form. By sending a continuously varying signal, the sender can detect how far "away" in milliseconds of propagation time the receiver is.)
Note that unlike an ultrasound ranging device (e.g. with a piezo to chirp and then detect the return), the sound is only propagating one way, with the other way carried over light.
At 90% of the speed of sound in the forward direction, the sound waves aren't going much faster through the air than the platform (and sender / receiver), and sound will take about 10x longer to get there. (Tending towards infinity in the limit as you get closer to the singularity in your formula.)
In the other direction, at close to Mach 1.0 sound will take close to half the time. In theory you can make the propagation time arbitrarily short by going many times the speed of sound, but hearing anything over the sonic boom will be problematic. And if you get into relativistic speeds in air, you will have bigger problems (xkcd).
Connor's answer shows that the speed of the source and receiver relative to the air can cancel out for subsonic cases, so you hear the same frequency. But that's not at all the same thing as a constant speed of sound.
Unprotected humans will have a Bad Time at transonic speeds (just below Mach 1), so let's assume sender and receiver are androids, or that the humans left a computerized experiment set up and wisely got off. We're also neglecting that air having to go around things will have to speed up a bit, potentially going supersonic.
This is why planes not designed to break the sound barrier will also have a Bad Time as flows around parts of the aircraft become supersonic. One of the speed limits for normal passenger jets is a maximum Mach number of typically 0.85 or so for the plane as a whole, which keeps airflow over control surfaces and wing roots comfortably below Mach 1.0.
All these speeds are of course relative to the air, which might be moving relative to the ground (aka non-zero wind speed). Mach numbers are implicitly air speed, so yes, the air itself is a special frame of reference, relative to which true air speeds are measured.