In any case, the headset needs to be greased......that notwithstanding..
I don't think I understand for sure what the problem is.
Put the bike on a stand, front wheel off the ground. Moving the bars from lock to lock, you should feel NO resistance in the headset. Maybe a bit stiff when turning to the right due to cables, but it's not hard to differentiate one 'feel' from the other.
Where the bottom bearing will first be felt is moving the bars either left or right of center. If there is any detent in the center, the bearing is at the very least in need of grease.
As I noted before, it's much easier to feel the bearing 'snick' when it's moved if the front wheel and the bars are off the bike. That's another indication of 'needs grease' (at least).
If it's just 'stiff' all over, and no 'snick' or detent, maybe you'll be lucky and get by with cleaning/regreasing.
What grease? The best you can get. Don't know if '20yr old tub of' fits that description or not.;) It needs to be waterproof and tough. What's currently sold as chassis/multipurpose (black moly-disul) grease seems useless for the task imo.
It's the 'when wheeling the bike..' part I wonder about.
Oh..try this. Lever the front end off the ground using the kickstand as the fulcrum. The bars should drop from side to side when centered by hand, then let go. It should drop pretty hard to the LH side, less to the RH side (again, cuz of the cables).
Easy project to take apart, clean and grease. Tougher to remove the bottom bearing. Some have effectively used a punch/drift and hammer to remove it. That was someone that worked with bearings for a living as I recall. He likely used a few tricks of his trade I'd guess.
It's hard to remove the bearing with a puller, cuz there is no purchase for the fingers of the puller.
It's common to press the stem OUT of the bottom clamp, pressing the bearing up the shaft in the process. I'd prefer to not mess with the stem if I didn't have to.
I cut mine off with an air cutoff tool. Worked good. Put new bearing back on with a piece of 1 1/4" pipe, capped on one end and a 2 1/5# sledge. One moderate tap and it was seated (on a stem that had been in the freezer overnight).
My local bike shop told me $45 or so to take it off.
Now then...don't you feel a lot better???