:I wanted you to explain it to me so I could better understand what is it you don't understand.  What you've 'explained' to me above is incorrect and demonstrates that you don't understand impedance or the relationship between phasors and the s-domain or even the relationship between the s-domain and the time domain.  I actually thought I had expained some of this rather well in my lengthy post above.  However, either I failed to do so or you didn't bother to try to understand what I wrote.  I'll close by individually addressing some of the things you said above: 

 ::"phasors require a constant frequency, which doesn't work for any arbitary wavefunction."  Yes, a phasor is associated with a particular frequency but that in no way implies that phasors cannot be used for arbitrary waveforms. 
 ::"And setting omega = to 0 after time 0 gives you an impedance of infinity, which is obviously incorrect except at t=∞".  This sentence is not even wrong - it is meaningless. 
 ::"And the impedance for the capacitor does depend on the input, however it doesn't depend on the s-domain version of the input (which makes s-domain impedance more useful)."  Again, not even wrong.  At best, this statement can be described as confused. 
 ::"As for the significance of s, I'm not an expert in laplace tranforms, but s is a parameter that can be related to the frequency of the input."  How can this be?  As you've already correctly stated, 1/sC is independent of the input signal. 
 ::"One can substitute jω for s and find the frequency response."  This is the closest you came to demonstrating some understanding of this subject.  It's not quite a correct statement but it's not far off. 
 ::"The s-domain is "better" than phasors for handling the unit impulse, because phasors require a constant frequency and laplace transformations don't."    Yeah, you've said this before no matter how many times you say it, the fact that it is wrong is not going to change. 

 :Now that I understand the depth of you misunderstanding, I'll be reviewing your edits much more closely.