The spindle is shown here fully disassembled.
Removing the spindle requires the removal of the end gear, lock nuts, inner race and sleeve. The back gears are a sliding fit on the shaft but the sleeve and bearing race are press fits.
Removing the end gear and lock nuts is the very first step.
You should be able to see that by pushing the back gear from the chuck end, this will apply pressure to the sleeve. This in turn will push against the inner bearing race.
The gear-end bearings showed absolutely no signs of wear. Impossible to distinguish from new bearings. The chuck end bearings showed very slight wear, probably due to some contamination. Definitely not sufficient to justify a new bearing. It has at least another 27 years remaining life.