Huge thanks to James III. With the detailed build description and videos the build was straight forward.
I decided to try going with a cheap new electric sewing machine. Found this one as a warehouse deal on Amazon.
It turned out to be a very good match for this project. The height perfectly matches, the hoop bottom moves less than 1mm above the needle plate.
Internally it's driven by belts run by a 12v motor. I replaced the motor by a Nema17 which is also driven by the grbl shield with a normal stepper driver. The stepper motor mount is custom modeled, as well as the pulley matching the original belts - thanks to GRITLabs Pulley Generator.