You are right except LH acts elsewhere also.
FSH is only responsable for the production of sperm.
LH has a half life of 30 mins while FSH has a half life of afew hours (6 or 12?). LH will vary so much throughout the day that it isnt so reliable to use as a follow up for how much you are being shutdown, instead FSH is more reliable, why FSH gets shut down from testosterone production, I do not know.