How to calculate bandwidth of a wire, "air" or other channels?
First of all note that bandwidth may have multiple (but similar) definitions. A practically accepted one defines the bandwidth of a baseband lowpass channel (indeed a filter as an LTI system) as the frequency at which the magnitude of its frequency response falls to $1/\sqrt{2}$ of the value at the frequency $f=0$. This point is conventionally called as the $-3$ dB point, or as the cutoff frequency of the channel (filter).

A physical conducting wire can be electrically modeled (depending on the frequency of application) as an LTI system with relevant parameters of $R$, $L$, and $C$, according to its resistance, inductance and capacitance respectively. Such a model therefore will define an LTI filter with a frequency response magnitude which will have theoretical $-3$ dB point(s). In practice the bandwidth of the filter could indeed be measured by a set up similar to what you have defined.

The air as a channel means either the empty space (vacuum) or the atmosphere. The former case ideally has an infinite electromagnetic bandwidth which is practically limited by the ability to create EM waves and the physical space's intrinsinc space-time and energy-mass characterization. The latter has a much narrower bandwidth determined by the electromagnetic characterisation of atmospheric air composition and its different layers, furthermore the sperical shape of Earth and the conducting soil also affects the channel characterisation. It's well known that the channel bandwidth of atmosphere depends on frequency, especially the ionosphere layer.