Couldn't the skimmer be mounted externally and read the stripe as the motor pulls it in?

My understanding is that the motor pulls the card in using random speeds which vary as the card is getting sucked in (start sucking, slow, fast, slow, fast, etc, fully sucked in). The machine knows how to account for the variance, but the skimmer doesn't (in general skimmer's assume a constant swipe speed). You can kind of see it cause the machine does the same thing when spitting out your card. It would be more convenient to spit your card out fast like a transport ticket, but then your card could be skimmed on the way out.

If variable speed would prevent the reader from working, how do the manual swipe readers work?

Cards are usually swiped in a somewhat smooth motion. Even then, it's easy to fail the motion, and you often need to swipe again.

The anti-skimmer moves the card very fast in short bursts, so while it may seem just a bit jittery (averaged out), the skimmer will end skipping whole pieces of band or losing sync.

Maybe they could use small rubber rollers to detect velocity and sync appropriately, like in a mouse.

That does not deter skimmers in any measurable sense. Most of the ATMs in Europe pull your card in by motorized mechanism and still skimmers are occasionally found on them.