[audio cassette player] Are both or only one "axis" supposed to drive the tape?
(piefed.cdn.blahaj.zone)
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.