How do you remove the chuck on a Guardian table drill press BDM-58-5S? 5/8" CHUCK.

SOURCE: Replace drill chuck on 23 year old Craftsman 8" drill press

Yes it does have a Morse taper and you can remove it by gently hitting it with a rubber or nylon mallet.

SOURCE: the chuck has come off my delta desktop drill

clean the spindle and the inside of the chuck with easy off oven cleaner. get it all very clean and then be sure to remove all of the debris. pound the chuck onto the spindle with either a rubber mallet or put a block of wood under the chuck and use a regular hammer. DO NOT hit the chuck directly with a steel hammer.

SOURCE: Dewalt Drill DC728. Chuck came out of drill body

If you find that the end of the screw inside the chuck is actually broke then you have a gear problem inside your drill. This is actually a very common problem. Sometimes people push too hard on their drills and the gears inside wear out and break. This prevent the motor from wearing out, but, the gears have to be replaced.
If it is the gears and not the chuck. you broke the minimium of the front gear or gears inside your drill. YOU CANNNOT repair gears. they need to be replaced. you may be able to find some off brand gears to work but, you run the risk that they are not as strong as your original ones, or may not be exactly grooved correctly.
so, if replacing your chuck does not work. find replacement gears. IF you cannot find the gears from the manufacturer see about replacing your drill.
If the chuck is still good, keep it. If you buy the same model, keep your old drill for replacement parts, ie, switch, armature, cord, etc.
hope this helps somewhat.

SOURCE: Craftsman Drill Press chuck problem

It is unless there is a tang on the end. What you need to do is thoroughly clean the parts with something like brake clean to remove all the oil and then wipe with a clean paper towel. What I do it put the arbor into the drill press or the chuck onto the arbor and then put a 2x4 on the table and press the parts together or if you have enough room fully retract the jaws and use the block of wood to "tap" the parts to seat them. Let me know if you need more assistance

Testimonial: "Thanks rowdy45176. Your solution was spot-on. I cleaned both sides and them , with the chuck jaws fully retracted, tapped it with a plastic mallet"

SOURCE: Craftsman 10" bench drill press chuck wobbles

If this is a new tool I would take it back and get it replaced. The only other way to eliminate this problem is to determine if it is the spindle or the chuck and replace the defective unit. Warranty works better.