LD does not significantly affect longitudinal stablity, only the balance of forces in level flight. ....
In fact, the tailplane, like the wing, has received the same finite increase in positive alpha and this would be the same with or without LD. I suspect the real reason for LD is to minimise trim drag in the cruise
That's pretty close
The tail receives the same increase in positive alpha from the gust as does the wing, but the effective increase in tail AoA is reduced because the wing generates more tail downwash as a result of its own AoA increase. As you say, this is independent of LD, which basically affects the forces in level flight.
Typically, the wing root may be set at 3 or 4 degrees nose up relative to the fuselage datum - this gives a more or less level cabin floor in cruise. The downwash generated at the tail by this wing AoA is
roughly enough to generate the tail download needed to trim the aircraft with no elevator deflection and the tail set parallel to the fuselage datum. So you naturally get LD if you design the aircraft geometry to match cruise conditions.
Obviously these are ballpark numbers and the details will depend on the particular aircraft design and the actual CG position (I took a mid-CG), but the general principle is valid I think.