Wouldn’t using projected area tell you the true values for lift and
No. For a given airspeed and angle-of-attack, a wing makes the same amount of lift, and drag, regardless of the bank angle. The "projected area" is irrelevant.
Otherwise, all you'd have to do is give a wing lots of dihedral, and it would automatically tend to roll toward wings-level whenever it was banked, regardless of whether there was any sideslip or not. Because the lower wing would have more "projected area" than the higher wing.
That's not how it works. The stabilizing effect of dihedral is intimately connected with sideslip, which increases the angle-of-attack of the low ("upwind") wing and decreases the angle-of-attack of the high ("downwind") wing. Take away the sideslip, and the stabilizing effect of dihedral vanishes.
That's why increasing the vertical fin area can promote spiral instability-- because it reduces the tendency to sideslip while turning.
The tendency to sideslip while turning may be so slight as to be difficult to perceive, but it always there.
If you don't believe that the stabilizing effect of dihedral is dependent on sideslip-- if you are instead a believer in the (faulty) "projected area" theory-- then try using the rudder to command a slight amount of skid (rudder toward low wingtip, ball toward high wingtip), rather than slip, while banked and turning, while keeping the ailerons centered. No matter how much dihedral your plane may have, it won't roll the plane back to wings-level if you are not allowing the plane to sideslip at least a little bit as it turns.