High-sensitivity loudspeakers in general have relatively simple (few parts) crossovers. Low-sensitivity designs are often of low sensitivity because the crossover uses resistors to reduce the sensitivity of one driver (usually the tweeter) to match the output of a higher-sensitivity driver, thereby sacrificing overall loudspeaker sensitivity in the pursuit of flat frequency response. Such designs therefore require more power to reach a given SPL, some power dissipated as heat by the resistors in the x/o.
But there are some low-sensitivity loudspeakers---planar-magnetics in particular---which have very simple crossovers. The Eminent Technology LFT-8b, for instance, has simple 1st-order high-pass and low-pass filters (requiring only a few parts to implement), yet is a very low 84dB in sensitivity. Maggies too have relatively simple crossovers, yet require gobs of current. So it is not necessarily the crossover which is "eating" power.

