I do think this is true, however, I'm not sure about how valid it is if you have manufacturing defects. After all, there is a fatigue limit. The spindle surface is brittle. As few as one high stress cycle could cause a surface crack to propogate and invalidate the prior fatigue life history.

Note, in Dawes-Man's bottom-most photo, there is a rather large "feature" which he has not yet polished out. In fact, the polishing has served to highlight it. It is on the right side of the shaft, almost dead center in the photo. Note also, there is a circumferential groove just slightly above that. In a brittle, case-hardened part, either or both of these could be a site at which a crack develops from fatigue.

I do like the idea of polishing the surface to minimize defects. This is the type of manufacturing processing that raises the price of the parts, so it wasn't done.

Do we know if these spindles were machined or forged (or both)?

Shot peening is also used to effectively strengthen parts. It induces a residual compressive stress in the surface, so the effective tensile strength is higher than before peening. Again, this costs money, so it wasn't done.