So everything in a drum works together, the heads, the air inside and the shells, but if we look a bit further around the drum, and provided the number of solutions and devices proposed for « suspension » we will soon find that those have indeed a huge effect on the sound produced.
The figure below shows the ratio between a force applied at the center of a drum, and the vibration output, versus frequency (known as a transfer function, or local inertance).
The green curves shows the vibration of the head when the drum is fully suspended (i.e. attached to elastic bands resulting in very low frequency suspension mode).
The black curve is the same drum, and same measurement now made on a snare stand.
You can observe the lowest fundamental being almost killed, the second unaffected and the third and fourth looking shifted in frequency and damped. In higher frequency, this difference is not that noteable.
Bottomline : a drum struck on a stand is a multi stage coupled mass/spring system – meaning ultimately that the « suspension frequency » (the one at which the drum is showing only rigid body motion) must depend on the lowest head frequency and thus its tuning.
Probably often achieved say for a 8 or 10 inch tom tom with proper « isolation » system, certainly questionable for a snare on its stand and bigger tom toms. Beyond « mounts » type system where everyone can conceive elastic deformation of the material used, keep in mind that a stand itself, or an L rod on which a drum is cantilevered too, can constitute a very good decoupling system : try mounting your toms on top or on the bottom of the rod to change the « free length » and therefore first frequency of the system (with the 4th power of L)