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I looked at the vid several times. And only notice the slightest amount of side to side of the tire’s edge being out of line.
Not enough in my opinion to raise a red flag here
I would put it back on the lift and try this!
With the bike as steady as could possibly be. And this is most crucial.
Then attain a piece of white chalk. Now locate the absolute center of the tire. Run the bike at the slowest speed and track a chalk line.
This is the part that matters.
The center. Nit the edge
Pull the wheel off. There is a line around the tire about 1/2" outside of the rim. It should be even with the rim all the way around. I know you said you checked the bead but double check anyway. Now pull out your bearings and inspect them for wear. After you repack them check your wheel runout (see procedure in manual). Before you reassemble the wheel on the swingarm, check your axle to make sure it isn't warped. Put everything back together but don't put the belt over the pulley. Tighten it down and give the wheel a spin. If it's still wobbly, I'd say you have a bad tire or the rim is warped. You'll have to pull the tire to check the rim for warping. I'd also make sure the pulley is tight on the wheel.
When you put it all back together, check the axle adjustment carefully so it runs true. Keep us posted on the outcome.
How does this effect your ride experience? Does your rear end (***) feel like its sliding side to side. Does it scare you like you’re gonna crash? Or does it feel warm and cozy. Every little bit helps.
From your description it sounds more like the steering neck bearings are over tight or seizing up/ flat spotted. Have you greased the neck bearings or checked the fall away?
Your pulley looks like it's tracking straight, as does your wheel when focused on it right above your pulley. Looks like a bad tire. Old rubber that should be replaced anyway.
When you replace your tire you'll have a chance to check your bearings, spacing, and runout. You'll need a shop to do it or invest in some tools. If you're mechanically inclined tools are an investment that goes well with having an old FXR.
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