Can a motorcycle speedometer be used in a Barth instead of a broken original??

Short answer: maybe

Long answer:

It will depend on several factors:

First and foremost, is the direction of rotation the same? Must be.

Most motorcycle speedos I've seen have a square drive center cable with an outside threaded sleeve to secure the cable to the speedo. The Barth cable end may or not fit, possibily requiring some modification.

The calibration may not be the same. The cycle speedo may run faster or slower depending on the rpm of the cycle cable at a given mph (oops, sorry, kmph… been in this backward country too long) The Barth cable runs from a speedo gear in the transmission. There is some room for adjustment here, there are gears available with different # of teeth, hence different ratios. Also if you are clever, it is sometimes possible to adjust/modify the speedo head internally to get the correct indicated speed. (The gap in the magnetic coupling can be tweaked, spring tension on pointer also)

If the stars are right you get a cable that plugs right in and shows the correct speed. If not, see above.

I got tired of my Barth speedo, indicated mph varied 7 mph depending on cold or warm, etc. I replaced with a Speedhut GPS speedo. It records speed, distance, trip, 1/4 mile, and maximum speed… (mine got to 80, honest officer I didn't think it would go that fast, maybe I should reset that one) Excellent but pricey.

good luck