Bandwidth required for an arbitrary digital signal
Why is it seemingly universally recommended that you use the rise time to find the spectral content of a digital signal and not the Fourier Series representation?

Because if you consider waveforms with different rise and fall times, they'll have different Fourier series (or Fourier transform) representations.
The width of the spectrum in the frequency domain will drop as the rise and fall times increase.

I can find the rise time for a signal easily with an oscilloscope, however, once I run that signal though anything, a longer cable, a coupler, an amplifier, a power splitter, an IC, won't the component affect the rise time of the signal?

It depends on whether you've chosen those components with enough bandwidth to support the rise time you want to have for your signal.

How do I know the bandwidth requirements of the signal after its gone through some component or device? How am I supposed to know the bandwidth requirement at any point along the signal’s path without directly measuring it after each component or device?

Practically, it doesn't matter what the rise and fall time are at the signal source. More important is what rise and fall time you need to have for your receiver to work reliably. Then design your signal path with enough bandwidth to be sure that the rise and fall times don't get too slow before the signal gets to the receiver.