How do I wire a modern 240V 3 prong range to a 250V 50A 3 prong outlet?
That white wire can just...go
Since your new range doesn't use a neutral, and this circuit appears to be run in conduit, I would turn off the feeder to the subpanel box at the main panel, make sure the power's off in the sub, then unhook the white wire headed rangeward from the neutral bar of the subpanel and pull it out as a starting point.  While this may sound odd at the moment, it's actually a rather sensible move since you don't have separate neutral and ground in the feeder, thus we need to pick one or the other to provide at the subpanel, and since the other load has no use for neutral, we pick ground.
Of course, if you are planning to consolidate the panels, then the white wire can stay since it'll become a proper isolated neutral in that case.  You'll want to cap it off at the cooktop junction box in the meantime, though.
That subpanel needs to go, though, too
However, once that white wire's gone from the equation, the challenge begins, as sadly, that subpanel is a Zinsco, with all the relatively checkered failure history attendant to that fact.  Sadly, the fact it's recessed is going to make this a tricky (albeit rather necessary!) job, especially considering that retrofit interiors aren't made for small panels like yours.  You'll want to ignore the bonded bar and fit a ground bar to the new panel, though, landing the bare ground wires on the incoming feeder cable and outgoing 10/2 NM cable on the ground bar and leaving the bonded bar empty as a cue that neutral is not available at this panel.
Once all that's out of the way
Once you get all that taken care of, then the green wire from the range can just land on a small mechanical lug (Brumall 6T or equivalent 4-14 foot lug with a #10 hole in it) connected to the box by a 10-32 self-tapping screw (Garvin GSST).  From there, the conduit continues the equipment grounding path to the subpanel, which then is grounded via the bare outer braid of the SEU cable used to feed it.