Demodulating signal by passing incoming carrier wave through low pass filter as opposed to multiplying by carrier wave?
What you're missing is that multiplying a signal by a sine wave changes the frequency at which the signal appears.  Running a signal through a linear time-invariant filter (what you mean above when you just say &quot;filter&quot;) can emphasize or de-emphasize the signal content at certain frequencies, but it cannot move the signal in frequency.
The classic explanation for this is to consider a signal that's a single sine wave at some frequency $f_{RF}$.  So, $x(t) = \cos(2 \pi f_{RF} t)$.  Put it through a low-pass filter and it's just gone.  However, if you multiply that signal by a local oscillator, you generate signals at the sum and difference of the signal frequency and the carrier frequency.  Let your local oscillator signal be $x_{lo}(t) = \cos(2 \pi f_{LO} t)$.
Multiply them together, and you get $$x_{IF}(t) = x_{RF}(t) x_{LO}(t) = \cos(2 \pi f_{RF} t) \cdot \cos(2 \pi f_{LO} t)$$
Now it's just trigonometry: $$x_{IF}(t) = \frac{1}{2}\left(\cos \left(2 \pi f_{RF} - f_{LO} \right) t + \cos \left(2 \pi f_{LO} + f_{RF} \right) t \right)$$
Now you have something useful for a radio receiver: the signal at $f_{RF} - f_{LO}$.  The only problem is that you have the other signal, too.  So now you run that through a low-pass filter -- and the result is your desired signal.