I don't have a cassette player around, but I'd think that it'd be extremely difficult to have a motor drive the unspooling side concurrently, because the ratio at which you'd need to drive the two gears would change over the span of the tape due to the size of each spooled-up tape radius changing. And tapes aren't standardized in length, so it's not like the tape player—even if it had a way to know where it was in a tape when you inserted it, which it doesn't—would really have a way of knowing that ratio.

Only one of the spindles will be active at a time, the one collecting the tape from the other, if the player has an auto reverse mechanism it'll switch between them, if not, it'll just be the one

If you were curious, there's also a capstan and pinch roller that you won't be able to see that is responsible for holding the tape onto the tape head and moving it along, so it's not tension from the collecting spindle that is dragging the tape through everything.

Yes. But you're holding it upside down. Normally the right one pulls the tape through the head. The logic goes that it reduces the number of components in the build, and if there is a problem you don't want the trailing spool continuing to push tape into the mechanism.

In an auto reverse deck the active one will stop and the idle one will start. That pulls the tape through in the opposite direction to read the second track. In a normal deck you'd flip the tape around because it only pulls in one direction.

Edited: I looked at the pic and saw that your tape goes in upside down.

I always called them spindles or gears.

Anyway, you can test this by running it without a cassette in it. As others have pointed out, one side pulls and the other is idle.

I kinda miss tapes. K-Pop groups still release them. They are not cheap.

If only one drives the take it won't rewind. For rewinding the A side you have flip the tape and go full forward on the other side. If both sides have "traction" you can reverse and go firward freely.

yes, it's used to rewind, not play.

I had one where the feeding spindle was smooth and fixed.

The only reasons to have the feeding spindle grip the gear and move is to rewind the tape and/or collect the tape in auto-reversing decks.

Many portable players didn't have rewind/auto-reverse fir cost reasons, so the feeding spindle would be just a fixed pin.

So, why not get rid of it? Well, to prevent the feeding roll from wobbling and causing jams or damage to the cassete or tape.

It's been a long time since I used one of these. But back in the day I had multiple 'walkman' style cassette players (one Sony, one Pioneer, one JVC, and over it two others I think). And from what I recall, they almost all only drove from one of the two spindles.