Hello Yves,
Well, it has been a while since I looked into the inerts of a speedo for the last time and I also do not remember all details
but I do remember some.
First of all, the brass gear you are indicating with your red-white arrow does indeed look a bit worn.
However, I do not think it is oriented incorrectly.
If you look at the orientation of the pinion's teeth at the red-white arrow-tip, the 8 o'clock position, you realize that the teeth are oriented from upper left to lower right. Now, if you turn the wheel by 180° towards the perpendicular worm gear towards the 3 o'clock poition, the teeth of the brass pinion are oriented the other way around: upper-right to lower left, and that is the correct orientation for the worm gear drive. Maybe the distance between worm drive and brass pinion isn't absolutely correctly adjusted, and to me it also seems that there is some grease missing...
If I see the orientation of your speedo correctly, the dial disk is up at 12 o'clock on your picture and the bulbs go in from 6 o'clock. Have you checked what this perpendicular metal shaft does? It appears to me as if that is the drive for the reset knob rather than something that constantly turns while the speedo is working. Have you checked that? You can use a power drill, with a drill covered with a bit of tape. Pushing that into the flange for the speedo cable and running the power drill counterclockwise
should rev the speedo up to about 50 mph.
Then you can see what this mechanism you are showing really does.
Personally I think it is only the reset mechanism for the tripcounter or so.
It is more likely that one of the plastic gears directly attached to the odometer or tripcounter shafts is cracked or loose on its shaft; the very common failure of these mechanisms.
Personally, I would first confront the initial shop with the failure and see how they react,
then you can still send it anywhere else if you don't trust them anymore.
Please keep us updated.
Personally, I have made the experience after getting several speedos repaired with several specialized shops, our instruments are ++ 50 years old and still repairing them doesn't turn them into "new" instruments anymore. There are quite some interts that will always remain "old".
Best,
Achim