Underneath you can see me starting up with a cold bike, pulling the clutch lever, shifting from neutral to 1st gear, back to neutral, then 2nd gear, then neutral, then 1st, then doing some brake testing,
Looks completely normal to me up to 1'25". The way the wheel accelerates with the clutch in is 'pretty normal', can't tell that there is anything wrong with that.
I mean, OK, maybe it clonks more than it should and is more 'draggy' than you'd like, but that would be consistent with a bike that's been standing around a long time, which I get the impression it has, and maybe the oil is a bit thick and not changed recently.
and then proceed to 2nd, 3rd and so on to 6th gear, and then down shifting again to neutral (in this process, you can hear a sound indicating a "rough" move from neutral to 1st, as forementioned; it's important to note that this is only happening when the bike's cold).
The clutch actuation sound very weird at this point. I can't really tell if it is or is not changing, but the way the lever jerks seems odd.
When it comes to "the bike won't slip into next gear after shifting", I'm sorry about the unprecise language. What I mean is that the bike won't slip into gear immediately when I try to shift, but slips into gear with a 'clonk' when I let go of the clutch lever and it's about halfway out
That may be because of some strange clutch movement behaviour, that we just mentioned.
It is not abnormal for gears to not go in immediately, if the wheel is stationary and the dog clutch is not in the right place, you have to have the wheel turning before you know if that is really an issue.
All I would say is that up to 1'25", that bike looks to me ready to ride. Just get it on the road and give the clutch some good slipping as you pull off, loosen the friction surfaces up.
From your description, it seems to me you've not ridden this bike, and it has been standing for a while. Is that so? And then you've been checking it out, tested it on a paddock stand and started worrying about this.
TBH, engine sounds good, does mostly what it is meant to do. Time to get a crash helmet and gear on, and give it a run, get that clutch slipping to burn off the top layers of disc surface, they are designed to be slipped not sit static for months.
Then once you're persuaded it might all be over-thinking it, and it's actually working OK after all, change the oil as the friction discs will have shed their top layers into the oil.
Sounds to me like it's running nice, TBH.
PS: Just an extra thought. if you do ride it on the road and you're worried about whether the clutch might fail and it'll keep going with you on it, nah, the brakes are more powerful than the idling engine. If your clutch really does fail, just roll off the throttle and leave it closed, keep it in a high gear, and brake to stopped and stall the engine out. If you have presence of mind and enough time, it wouldn't hurt to flip the run switch to 'off' before you come to a stop. But not essential, just keep squeezing the brake. Be aware like that the bike will stop suddenly just at the end, be ready to stabilise it with your feet. I am saying this bit only from thinking through what would probably happen, can't honestly say that's ever happened to me.