I'm not seeing enough clutch wobble in the video that I would expect to influence turning finish at all. I don't think that is the source of your problem.
It looks like the material you are trying to turn is quite hard/tough. Is it a piece of unknown scrap? If so, replace it with a known piece of mild steel bought as such from a reputable supplier. Bits of unknown scrap often prove impossible to turn nicely.
Also, your indexable tool could be a source of trouble. It is sticking out of the toolpost too far for starters. And, hard to tell from the video, but it looks suspiciously like a 12mm tool shank, which would be setting the cutting tip more than 2mm too high above centre. Even a 10mm shank tool sits 20 thou or so above centre in a standard Myford 4-way tool post, which is made for old 3/8" tools, unless you mill down the toolpost base or the tool shank. Or you can use 8mm shank tools sat on packing strips of sheet metal built up to height. And do check the tip of that tool bit with a magnifying glass and make sure it is not chipped.
You could also try dropping the rpm by putting the belt on a medium speed headstock pulley and see if that helps.
And be aware that carbide tooling like that needs to take a decent depth of cut, say 5 thou or more. It often rubs and plays up if you try to take light finishing cuts with it.
Before you start fiddling with your headstock bearings, actually measure how much play you have in the headstock bearings. THis is best done by holding a long piece of stout bar in the chuck and pulling the end up and down. Measure movement of the spindle right behind the chuck with a dial indicator to get a reading. Dont adjust it if not excessive. There are threads on measuring and adjusting S7 headstock clearance on this site you will find with a search. Read them before proceeding.