A strobe wouldn't need to be synchronized to the spindle. Years and years ago we used a strobe tachometer to measure motor speeds in tape recorders and suchlike. You'd vary the strobe rate till the part appeared to stand still and then read off the RPM. The part would appear motionless at any exact multiple of the RPM so you needed to find the lowest or highest rate to get the RPM, it was easy since you knew the approximate rate before you started. If the rate was a little off the part would appear to revolve slowly.A strobe synchronized to the spindle could be useful. That way you could see what's really going on in there.
These things can probably still be found - cheap.
Edit: I guess not so cheap: http://www.ebay.com/itm/110986536613