Then with your right hand rotate the clutch shaft and use the shift lever to try each gear and neutral. You should be able to feel it go into each gear.
Did you do this? --^
I'll try and explain a bit by referring to your picture. The shift lever goes through the cases and is connected to the large arm on the left of the picture with the little spring. If you move the lever up with your foot the arm moves backwards rotating the shift drum CW one notch. That act brings the bike up one gear (2nd to 3rd for example). I think my mistake was telling you the shift drum "spins". It doesn't "spin" per se, it rotates in a ratcheting motion. Each ratchet is a different transmission position (neutral or a gear depending where you were to start).
The little pizza cutter wheel on the top right of the drum won't always be in that little pocket. Only when it's in that specific gear.
If you just start shifting gears you'll see what I mean. So do that. Shift gears. Now.

Also, watch the output shaft when you're in different gears. You'll see the gear reduction happening. Notice how fast you spin the clutch shaft vs how fast the output shaft is spinning. The higher the gear, the slower the output shaft spins in relation to the speed of the clutch shaft.