Verlet integration uses an object's change in position at time t to determine the velocity for t+1- it is typically used for particles not full rigid bodies.
I would think that rigid bodies -- either perfectly rigid or semi-rigid via springs & struts -- would not handle all that differently in verlet since force is explicit (typo on your part?) and a v/p estimate is available from the last position, albeit with a bit more phase lag than usual. Does the phase lag induce nasty oscillations or something?
I've played around a lot with Verlet, it is very powerful in certain cases...especially since you can just update the position of a mass directly (very easy to program, very mouse/touch input friendly). However, a rigid body physics engine is probably better for most game use cases.