OK, so if I understand right....when you turn the rod, you can see the keys move with the rod ?
If that is the case, then in fact it isn't the rod threads which are buggered and frozen - it's the fact that the rod is sorta fused to the key barrel(s).
Although this sounds horrible, it's actually perferable.
But as Saxoclese says...IF the screwdriver slot is trashed, then that puts you at a disadvantage. If it's really stripped so the screwdriver does not grip/hold any longer....that's not great. The only way to re-slot the end of the rod would be to actually saw into the post head:
We've probably all seen the filing in to the post and screw to get a bigger screwdriver or better grip too.
....this can be done, but is not advised because then you've permanently put a slot/scar into the post head...and it's not guaranteed to succeed even if you do manage to reslot the rod head.
But the issue here is not the threaded end, it now becomes freeing the rusty rod from the key barrels.
My suggestion (
IF there's still sufficient slot left to hold the screwdriver to the rod):
1) Hit the seams of each key (the ends of each barrel for each separate key) with the oil. Let it soak in overnight. Honestly, even the BEST penetrating oil is just NOT gonna really penetrate very deeply into the barrels.
2) Then go to the heat and quench method. No...you need something cold as all heck...thus the ice cube. Air won't cut it.
3) NOW...instead of concentrating your effort on turning the rod...we know (assume the rod threads are fsomewhat free in the post head, so we wanna concentrate on freeing the key barrels). Put the screwdriver into the rod slot and HOLD it there, don't try to turn it.
4) then pick a key and grab it and try rotating the key up and down while you are holding the screwdriver/rod stationary. You are hoping to get some movement in the key without taking the rod along with it.
See how that goes.
If it fails, or if the slots are so far gone you cannot get any screwdriver grip on the rod - then you are at the worst case scenario, as noted by others ....and you'll have to bring it to a tech to extract the rods and indeed the likelihood is some of the rods will not be salvageable.
But it's not a completely lost cause. Just sort of a not untypical pain in the butt.....
But this is why when I buy a project horn I always ask in advance if the screw heads move