The short spindle is either the wrong one or else it is jammed.
Also, all of these machines I have ever seen suffer the same malfunction which has to do with skipping at the end of the record or the automatic cycle not working right. The fault is due to a small lever device on the cycle gear. It and the gear itself used a lubricant which turns into an effective glue over the years and decades. The cycle gear has to be removed and its axle cleaned and lubricated with a touch of WD40, as does the little lever mechanism mounted on the cycle gear. Once these operate freely the automatic cycle will work fine and the arm won't skip.
Also, all of these machines I have ever seen suffer the same malfunction which has to do with skipping at the end of the record or the automatic cycle not working right. The fault is due to a small lever device on the cycle gear. It and the gear itself used a lubricant which turns into an effective glue over the years and decades. The cycle gear has to be removed and its axle cleaned and lubricated with a touch of WD40, as does the little lever mechanism mounted on the cycle gear. Once these operate freely the automatic cycle will work fine and the arm won't skip.