I sure do not wish to generate more work for you, X-roads, but have you inspected, serviced, replaced or done the "grab n' wiggle" test on the rear wheel?
When I did a pre-buy test ride on my 1981 R65, I thought I must have surely forgotten how to ride! The bike wobbled so that I felt as if I was going to fall over!
Quick check determined the rear wheel bearings were loose. Sale price went down. New bearings, races and a full inspection plus new lube cured the wobble. I'll skip the pre-load DIY ritual. The bike's now a straight line goer!
Easy to check. Give it a tweak and report back. Good luck. Monte