How does this AM Modulator work?
I forget the name of that one, but that circuit is an oscillator. (Is this a commonly accepted variant of the 3R, 3C phase-shift oscillator maybe?)

It becomes an AM modulator because its power supply is your input signal - so obviously its output Amplitude will be Modulated by the input signal.

The point of that LC network is to provide a 180 degree phase shift at the frequency you want to oscillator to work at.

The transistor in this circuit is acting as the amplifier in the circuit so yes you are correct that need to calculate its Rc &amp; Re resistor values to produce a gain which compensates for the attenuation in the LC stage at your frequency of interest. (You've labeled Rc &amp; Re backwards btw).