It is a feeling thing, you don't actually get more vertical space because most games don't change the vertical field of view with the aspect ratio. If you're playing on 16:9 and you go to 4:3 it should give you more vertical space. But the games I've tested the the vertical field of view stays the same. Vertically you see just as much as with 16:9, but horizontally you see less because the aspect ratio gets cropped horizontally. Same applies with 21:9 and 16:9. You get the same amount of vertical space with 21:9 as you do with 16:9, the actual difference is in horizontal space.

The difference you feel is just an illusion. Even in that video I agree that 4:3 really does feel like you're getting more vertical space. However the cropped sides break the illusion and show that you're not actually seeing more vertically. But it's a feeling thing so there's no wrong or right answer. If you feel like you're getting more vertical space from 16:9 then you're getting more from 16:9. I don't feel that so I appreciate the extra horizontal space.