Sad to say but to get the V-Belt off you will have to either cut it or pull the spindle out of the headstock. You can help yourself if you're willing to use one of the segmented V-Belts which is made up of links that attach to each other. You can make yourself a belt of the correct length and then wrap it around the pulley's and connect the ends. That way you don't have to take the spindle out which is not a particularly easy job on that lathe.
Here's an example of one of the segmented belts. I don't necessarily recommend that you get it from Amazon but it does show what it looks like.
Those belts are very effective and would work just fine in you application. I use them on my table saw and on my vertical mill and I've never had any trouble with them.
The reversing tumbler gear set that has you puzzled is removed as a unit. If you remove the clamp screw the whole assembly will pull out of the headstock. Once it's out you can access the individual gears and spindles. From the video it appears that grease has been used on those gears and that is not the recommended lube for those gears. They want oil. Grease attracts dirt and metal shavings which you don't want.