Before you write off the spindle in your drill press, I have a few suggestions based on experience:
1. Jacobs tapers, used to hold drill chucks onto spindles, are locking tapers. The threaded collar found on some drill chucks is redundant if the tapers on the spindle & in the chuck body make up properly.
2. Clean the male taper on the spindle & the female taper in the chuck body with solvent such as automotive brake parts cleaner and possibly some fine steel wool to remove any traces of 'compound' that might have been used to try to bond the chuck to the spindle. I would then check the fit of the chuck on the spindle using chalk lines drawn on the spindle taper. Draw four chalk lines running lengthwise on the taper at about 90 degrees to each other. Light chalk lines are what's needed. Put the chuck on the spindle, press it home (not hard enough to seat and lock it on the taper), and twist the chuck on the spindle thru a portion of a turn and back again. Carefully remove the chuck from the spindle, trying not to let the female taper inside the chuck slide or drag on the chalked lines. When the chuck is off the spindle, take a look at the chalk lines. These will show what kind of contact the chuck taper is making with the spindle taper. If you can get a tube of "Prussian Blue" (an oil based blue paste, not unlike an artist's oil paint, often sold in automotive parts stores or order from a supplier like MSC), do a "Blue check". Apply the Prussian Blue as one small dab on the male taper, and spread it to a thin hazy film using a piece of tissue paper and possibly a few drops of mineral spirits. After you have the hazy film of Prussian Blue on the male taper, set the chuck on the spindle working it to full seated depth, and 'wring it' on the spindle thru maybe 3/4 turn and back again. Remove the chuck from the spindle and see where the Prussian Blue was rubbed away. Either the chalk test or the "Bluing In" will tell you whether your spindle taper is not matching the taper in the chuck.
3. If the taper on the spindle is off by even a small amount, it will not let the chuck lock onto it.
4. Years ago, I saw a fellow doing a 'government job' (a job for something he had at home, not for the company). He had brought in a drill press spindle and said the male tapered end which mounted the chuck had been bent slightly. This fellow chucked the spindle in a 4 jaw lathe chuck and indicated off the straight journal immediately above the chuck taper. He had the spindle running dead true in the lathe chuck, and the indicator did show a slight bend. He used a dial indicator in the toolpost to set the angle of the chuck taper on the lathe's compound slide. Indicated off the unbent side of the chuck male taper. He then took a very light skim cut on the male taper and faced maybe 1/32" off the small end of the taper . The fit of the chuck was check with Prussian Blue, and a very small tweak of the compound was needed to bring it to the correct taper (leave the locking screws in the compound snugged and tap the compound lightly with a piece of brass). This combination of truing up the taper and taking some off the small end repaired that drill press spindle.
5. It's been my own experience that dill chuck bodies are hardened and ground, and the male spindle taper, while finish ground, is usually not so hard. What may have happed to your W-T drill is that someone ran too heavy a bit in it, or similarly put too much torque thru the tapered fit between the chuck and spindle. This 'broke the fit', and the spindle may have spun freely in the chuck taper. This could have caused a galling or scoring. A previous owner, having done that damage, may have cleaned up the spindle nose with emery cloth and thrown the Loctite (or maybe JB Weld if a real backwoods cobber) to it.
6. Making a new spindle may not be so easy a proposition if it has splines for the cone pulley aside from having to match the chuck's taper. Checking the taper's fit with bluing and dial indicator would be my first step. If the blue check does not show too bad a mismatch, you may get away with stoning the male taper to correct its fit in the chuck taper. Using small oil stones (India Medium Hard, and Arkansas Fine), with the drill press spindle turning under power in the drill press, you can dress the male taper slightly to correct the fit in the chuck. The India Medium Hard stone will remove small amounts of metal, and the Arkansas Hard Stone will feel like smooth marble to your touch- it will remove amounts of metal hardly measurable. Start with the India Hard stone, working it on about a 45 degree angle to the centerline of the spindle, stoning the areas where the Prussian Blue was rubbed off (these contact areas may appear as bright metal spots). Stone lightly with the spindle turning, clean with solvent and check the fit with Prussian Blue. A little stoning, frequent contact checks with the Prussian Blue, and a taper can be tweaked that last little bit. The Arkansas Stone is used when you are getting close to a locking fit.