Watched the video, but didn't tell us much. A couple of questions:
1. Did you find and reinstall the steel ball on the left end of the lifter rod? The lifter rod is the long piece that traverses the engine.
2. Engine oil is an important aspect of the clutch in these bikes. Have you drained and replaced the oil, and if so, what oil did you use? (Calm down everyone, not trying to start a popcorn rest!)
3. You indicate you ordered a new clutch cable, but is it installed?
4. Left side cover, the corkscrew. Did you re-grease when you installed it? There's a grease nipple beside it on the outer side that you can use in the future. Upon reassembly, grease it by hand and install. The slot in that corkscrew is used to adjust the travel of the lifter rod (moving left to right thru the engine). Imagine it not being adjusted fully and not "pushing" the plates apart far enough.
False neutrals- these can be from wear on the dogs and gears in the transmission. But, they can also be from an improperly adjusted clutch where the engagement/disengagement is not complete. Since you indicate that the bike gets "stuck" in a higher gear, I suspect you have both an improperly adjusted cable AND some wear on the clutch plates themselves.
If this bike sat for quite a while, it is very common for the plates to become a bit stuck together. On the right side, there's a cover that has the clutch basket, and alternating steel/friction discs. These are held in place with clutch springs. The plates need to fully separate when the clutch cable is pulled, then "squeeze" together fully to engage the gear. So think thru that process, and you can quickly understand the nuances of a proper clutch adjustment.
TwoTired just posted a response for another member on the full aspects of the clutch on this model, and it's worth the read if you can locate it. Post back your answers to the questions above, and now that everything is cleaned up on that cover, we can start to diagnose the pieces.
Clutch adjustments are Black Magic. It is extremely tedious to get right the first time you undertake it, especially with little experience. Be patient. Dave or TT will chime in and give you the step-by-step on how to adjust it properly and accurately. Then we can decide if you need replacement of any items.
By the way, if your lifter rod is shot, (looked awfully rusty in the picture, did it come clean?) PM me. I've got a brand new stainless one I made and you're welcome to the extra one (I made 2). I don't have the steel ball bearing though...