Take-up detail.

This view shows the small collar on the take-up spindle to limit end-float.  Washers could be used as an alternative but these are somewhat fiddly.  Collars can be obtained ready made but if you have a lathe one can be made in a few minutes. Another modification is the primary drive wheel.  I've used brass for this.  I'm sure you can be excused if you think it will slip, I can assure you it will not!  This principle has been used on rim-drive record players for many years and doesn't rely on spring pressure for driving contact.  The relative positions of the wheels result in increased grip as the load increases.  However, the surfaces of both brass drive wheel and the idler must be completely free from oil or grease.  Clean both with methylated sprit or isopropyl alcohol before assembly.