That's just not true. Although screens and radios definitely suck a lot of power, having the CPU handle traffic from all the connectivity better translates to better battery (especially in standby states). Also, the modems are usually built into the main SoC (i.e. CPU) these days, so node decreases translates to a more efficient baseband modem too.
Little of your phone battery goes to the CPU. It's almost all screen and radios.