Yes, that's the Delta-Grip chuck spindle, and your drill press is roughly from 1936 or 1937. Somebody removed the Delta-Grip chuck body, probably because it wasn't working. Those chucks had a relatively-soft steel sleeve and were all too easily deformed by the chuck jaws, after which they wouldn't run true anymore.
As a result, they replaced that chuck and spindle style with a standard #33JT nose spindle and a #33JT keyed Jacobs chuck.
Here's what your chuck was originally, on the left, and the Delta #970 replacement spindle with #33JT nose on the right:
As I mentioned in the previous post, the thread on your spindle may not be exactly the same as a 1/2"-20 LH thread, but in any case, it's too long to mount a threaded chuck on it, especially with the extra stub at the end.
Cat. No. 970 Delta spindles with the 33JT nose come up on eBay quite often, usually with a chuck included. That would be your best bet IMO.
You can easily remove your existing spindle. Just lower the quill until you can see the collar at the top of it in the opening in the head casting. Lock the quill. Loosen the set-screw on the collar, and the spindle will slide out of the quill. However, before sliding the spindle out, lift the collar up, and if you see a dimple in the spindle (from the set-screw point), remove the whole quill assembly instead, so you can get to the dimple on the spindle to clean it up with a file before sliding the spindle out. Otherwise, that dimple can make it hard to slide slide the spindle through the bearing bores in the quill.