The difference between Pitch and FPA certainly gives a rough guide to AoA, particularly changes of AoA. As has been said by others, it should be correct in still air, but becomes increasingly inaccurate as the headwind or tailwind component increases. That's because FPA is in relation to the earth, not the airmass. (So vertical movement of the airmass, particularly a microburst, will also cause inaccuracy.)
The wind-component problem is perhaps best illustrated by considering the extreme case where headwind equals TAS, in which case any rate of climb or descent would produce (unreadable) FPAs of +90 and -90 respectively.