If you transfer a character from an account that's already received the level 11 overseer achievement, they are reset to level 1 overseer on the new account. Once you re-level to overseer 11, they receive no achievement, no quests and no apparent way of obtaining them. Adding the level 11 overseer quest to the status vendor should solve the problem.
Did you try creating a new toon on the new account? I've noticed doing that on an account that's achieved L11/S2 overseers results in the new toon getting the reward/announcement dialog each time, though of course they don't get the crate w/S2 starting quests anymore. My guess is the same thing would happen on your "new" account because the game somehow thinks you've already gotten it even though the S2 starter quests aren't "installed" in overseers* (which it should have detected when your moved toon hit L11,) but it's worth a shot if you haven't tried it, assuming you have an available slot. You can just delete the new toon after, whether it works or not. Regardless, the solution you mentioned, or any way of being able to obtain the starter quests at will after L11, is needed. * In my experience, this is the factor that determined whether I kept getting the crate -- until I unpacked it on a toon and the quests auto-installed. After that, no more crates (but a bunch of un-deletable, un-unpackable "no destroy"-flagged crates what will apparently be with some toons forever )
