The increased thrust component from gravity will offset the increase
in drag due to forward c.g. The trimmed speed should not change.

I'm not talking about drag.

This is a stability/control issue. By moving the CG, you are changing
the speed for which the aircraft is trimmed.

My understanding is that hang gliders use this technique by shifting
their bodies forward and aft. I've never flown one, so I can't say
from personal experience.

I've never noticed that leaning forward and back changed attitude or
airspeed.

Well, either go try it, or trust me. g Probably takes a small
plane before the effect is noticable. I've only done it in a 152. I
will occasionally demonstrate to a student how they can climb,
descend, and turn using only body shifts and opening and closing the
doors.