Mechanically, there just doesn't seem to be any reason the clutch won't engage and stay engaged.>

You may want to check the clutch draw for specs.
So . bring 12v to it with an amp gauge is series . You are looking for 3.8 to 4.2 draw for normal clutch coil draw. Don't have to run car for a coil draw test.

.. and , to see if your problem is controls , just back probe the clutch coil feed wires with a test lamp/volt meter and see if the feed drops out when the clutch does .. that means no power ..back to circuit feed checks.