The LC filter is the combination of a low-pass filter (the inductor) and a high-pass (the capacitor). It is the simplest (least number of poles) band-pass filter you can find, and the response is correspondingly simple. The more complex band-pass filters contain more elements but have a better response, like a wider passband and/or steeper roll at the stopband.
Consider the FM tuner, and imagine the two extreme casses:
1) Well away from the resonant frequency, you can assume the LC is a short. Therefore, the two transistors act like parallel diodes, and give no amplification. Only if you have a very close AM station you will catch some signal.
2) Right into resonance, the LC acts like an open circuit. The two transistors become a Darlington pair, giving a lot of gain. Also, the base of T2 acts as a rectifier. You can expect a strong signal at the input of the 386. The frequency modulation will take the signal somewhat away from resonance, modultaing the amplitude of the sound generated.
About the AM example, they are only concerned about the resonant frequency. I think it's assumed, but not stated, that Vin and Vout have input and output impedances (in series), and the filter is placed in between. The input and output impedances, as long as they are purely real (resistances) do not affect the frequency of resonance, but the quality factor of the oscillator filter. So if you are only concerned about computing the resonant frequency, these resistances can be added to the parallel resistance R. So for a simple computation of the resonant frequency, the circuit provided is enough.