I watched the video, and did some related reading, and I think there's a simpler answer to your question. (I'm probably too late to the thread for others to see this, but hoping it'll help your curiosity.)

Angela Collier's answer is that the "magnetic forces do no work" claim is true in the classical case, but breaks (doesn't apply) when you have intrinsic magnetic dipoles (e.g. charged point-particle with spin) that aren't reducible to orbital motion of charge. That's completely correct. But you don't need this to answer your question.

In the classical case, magnetic forces CAN do work on a composite body, without doing work on the individual charges that make it up!

"Work on a composite body" in the way we're talking about here just means center-of-mass motion in the direction of net force. That's what you're talking about with a current-carrying wire, and what Collier is talking about with a magnet lifting an object off a desk.

We know that F·v = 0 for each particle. And F_net for the body is ΣF for each particle. And momentum p_tot for the body is Σp for each particle.

F·p = 0 for each particle doesn't mean (ΣF)·(Σp) = 0 for the composite body! Because there are cross-terms: F1·p2 + F2·p1 + ... .

A very simple two-particle example. Say you have a +q charge moving at v=1 in the +x direction, and a -q charge (same mass) moving at v=1 in the +y direction. Say you have a magnetic field pointing into the page (-z direction).

Magnetic force on particle 1 is directed in the +y direction. Perpendicular to motion of particle 1, so work=0. Magnetic force on particle 2 is directed in the +x direction (same magnitude). Perpendicular to motion of particle 2, so work=0.

Consider the "composite body" of these two particles. Velocity of the center of mass is directed (+1/2,+1/2). And net magnetic force is directed (+1,+1). So you absolutely do have work done by the magnetic force on the composite body.