Stumped on understanding a Feynman lecture about force from wire on magnet

I think your analysis is all good, except for your statement that $\bf{B}$ is in the $y$-direction.

This is true only for points that are directly beneath the wire.

Most points on your coil are not directly beneath the wire, they are off to the side a bit. At such points, the magnetic field from the wire has a vertical component, which gives a net force in the $y$ direction when crossed with the current direction at those points.

(Of course, there is also a y $component$ of B at every point on the coil, but the vertical force this causes is exactly cancelled by an opposite force acting on the symmetric point.)

Dave, please DO NOT ACCEPT my answer.

Sounds like a strange request, he ? The answer you should accept is Paul G's since he posted it before mine.

I don't want to steal it, just make it maybe a bit clearer. I did it in comments, but it will be more visible here.

As Paul G wrote, because the field lines are circles and thus though the $y$ component of the $B$ field below the wire is the largest, there is a small $z$ component in the B field produced the wire at the position of the coils (which have finite radius), and the sign of this component is opposite on opposite sides of $y=0$ plane where the wire is. The current in the coils have opposite $x$ orientations on these opposite sides, so the relative sign with the $z$ component is the same so their cross products (which are in the $y$ direction !) add up to a net force in that direction.

Contrariwise, the $y$ component of B is the same on each side, and thus the (separately much larger !) cross products with the $x$ component of the current on each side of the plane exactly cancel each other by symmetry, so there is no net $z$ component to the force.

Again the net $x$ component of the force is zero. The $z$ component of the $B$ field do have cross products with the $y$ component of the current in the coils, but if you look at them carefully, you also find that they cancel by symmetry.