Unfortunately yes, you may have to replace both chuck and spindle.
They mate together with what's called a "self locking" taper- just enough angle to wedge themselves together. The "lock", however, relies on two clean, smooth mating faces, that also have to be damn near the exact same angle.
Any burr, ding, dirt or ridge can keep it from locking- and it's clear as day that both the spindle nose and the chuck recess are badly scarred.
If you can't find another spindle, it might be possible to have a competent machinist regrind the taper. I'd have to see it in person and measure to be able to say definitively yes or no.
The chuck is an easy replacement, grab a Jacobs of the appropriate size and matching taper, and once the spindle's fixed, whack 'er in place.
Actually, best route would be to buy the chuck (assuming the spindle has the meat for a regrind) and then use the chuck itself as a "master" to get the taper just exactly right.
Doc.