So I understand that Fourier analysis is extremely important in daily communication, next to it being pretty interesting fundamental math.

Now I am studying the math of fourier transforms and the applications to differential equations. It is pretty complex but I am starting to understand it a bit.

But what is so important about the fourier transform? Why is it so important to get a linear combination of sines/cosines of a certain function/signal? Why can't you do whatever you want in the time domain (or frequency domain)?
Sin waves of different frequencies are orthogonal.  This allows communication between many devices over a shared interface if they use different frequencies. By putting the functions in the frequency domain, this allows electrical engineers to easier design their systems.
One nice property is it converts differential equations to algebra since the derivative of each complex exponential is just iω times the same value, so taking the derivative amounts to multiplying every coefficient with iω.  Similarly integration amounts to dividing by jω.

Taking that further in dynamic systems with an input x of some sort and an output y you might have something like:

x = Ay'' + By' + Cy

Now what is the output y for a given x.  Solving in time domain requires integrating this equation.  Possible for some simple inputs, but gets complicated for others.  In the frequency domain (where x and y are now the Fourier transforms of input and output) this is just:

x = (-Aω^2 + C + Biω)y\

y = x•1/(-Aω^2 + C + Biω)

Now if you ultimately care about time domain this is really just pushing the complexity for weird inputs into doing the inverse transform at the end, but often the exact time domain representation is not essential to compute useful things.  For example if the purpose of designing this system is to isolate a particular frequency range (perhaps like the filter in your radio that selects just the one station you want), you can get all the critical bandwidth and attenuation information from this representation without caring too much about the full detail of the specific input or output signal.  So you don't even really have to do any transforms in that case, you care more about how the differential equation affects the transforms of inputs to get outputs.  Your dynamic system defined by a potentially high order differential equation (for example a circuit with a bunch of capacitors and inductors) is transformed to just a frequency dependent gain (and phase shift).

Similarly you can analyze circuits directly in the frequency domain by assigning capacitors and inductors a frequency dependent complex impedance at which point the analysis is just like a bunch of resistors instead of involving derivatives (although you do have to carry some extra symbolic ω's around unless its a fixed frequency system like AC power).