Incidentally, if you start with the differential form -- some function for input, plus derivatives of the output -- you can plug in a function and get a result directly. Well, with the small cost of still having to solve the differential equation.

But, that can be done one small step at a time, if nothing else.
Which is precisely what SPICE does in the transient analysis mode: implicit integration, to solve the differential equation, given stimuli (arbitrary sources) and some system (the circuit and model you've entered).

When you do AC analysis, it's a small signal steady state analysis. That is, nonlinearities are approximated as various linear gains between points. Everything is set based on the DC operating point analysis, so if you've set up the circuit for the wrong operating point. (This can happen, where in transient mode, there's a big spike in the first few time steps (microseconds, picoseconds, who knows..) as the circuit settles from an incorrectly calculated DC operating point, to its correct instantaneous level.)
The AC analysis transfer function need not be straightforward, but it can still be approximated with poles and zeroes; this is what the Transfer Function analysis does (I never found it too useful, myself; it's too hard to spot the dominant poles and zeroes.)
Tim