Passive band-pass filter conceptual analysis
Your idea is qualitatively right - one filter attenuates low frequencies and the other takes off some high frequencies from what's left. Unfortunately the rightmost filter loads the leftmost filter in a way which is frequency dependent. The result isn't the product of independent transfer functions, as you already knew.

As commented, one can reduce the loading effect by making the reactances of the rightmost filter much bigger than those in the leftmost filter, but it cannot be complete fix. Another idea is to insert a buffer amplifier in the middle. 

Both ideas are considered wasteful, when compared to accepting the interaction and designing the circuit as whole. Then one can set R2=0, R1 is what the signal source has, a physical R1 is added only if the signal source hasn't one. Finally there can be the load in parallel with C1. 

This solution leads to bell-like frequency response curve. The minimum attenuation is at the resonant frequency of the parallel LC circuit. At that frequency the currents through L and C are opposite and there's no voltage loss in R1 except what the possible load takes. 

You cannot get other curve forms with only one L and one C. That can be proven in circuit theory. With component value selection you can decide the center frequency and how wide the bell is, but you cannot make  the curve steeper without reducing the width of the bell.

BTW. the circuit was a revolutionary invention in the beginning of the radio era. It made possible radios to operate at certain frequencies. You can well experiment with it by making simulations, there's no need to calculate exact values beforehand.