Why doesn't source affect sound's speed in doppler's effect?

In the derivation of the doppler’s effect formula
$$f_{app} = ffrac{v ± v_{observer}}{v ± v_{source}}$$
​It is assumed that the speed of sound is unaffected by its source’s speed (i.e it’s the same relative to the ground whether the source’s speed is 0, 5m/s, …) (the source only affects its ‘release point’.)

How is this true? isn’t this similar to elementary relative motion questions like a bullet being shot from a moving police car at a thief (The speed at which the thief gets shot is the muzzle speed + car speed). Isn’t the bullet in this example analogous to the sound produced by the source in doppler’s effect?

Also, consider 2 people sitting a meter apart in a plane that is going at the speed of sound and one person decides to play a song from his phone, with this logic wouldn’t the other person not hear the sound as it will have the same velocity and therefore 0 relative velocity with respect to the other person?

My hypothesis is that when we say ‘speed of sound’ in a medium, we mean its speed relative to the medium(In this case air) that is carrying the sound. In the airplane example, the air inside the plane is ‘moving with’ everything else in the plane so sound will go at its normal speed relative to THAT air(since that’s the air that’s carrying it) and therefore it will reach the other person, but in the doppler’s effect, the sound is travelling through the ‘surrounding/open’ air which is at rest with respect to the ground and travels with respect to THAT air and therefore has the same speed regardless of the source’s speed. Is this right?