"I have an very old Kenmore top loader which after 20+ years still
does a great job."
You mean it still does a great job other than leaving large oily
stains on your clothes. What would you consider a lousy job?
Anyway, even though you did not provide a model number, here's a
generic suggestion for older Kenmores:
Remove the top cap on the agitator (probably screws off)
Remove the large nut under the cap
Pull the agitator straight up.
P.S. I've heard rumors of a set screw on the agitator, but I never had
one on either of my Kenmores.
P.P.S. get rid of you 20 YO machine and buy a front loader - Less
water, less detergent, less dry time (meaning less gas or electric
usage), gentler on clothes, etc. etc.