It's Not the chuck
Most of the drill press use a morse taper that the chuck slips on. BUT it must be true and clean, no marks,on the chuck or the shaft.
Most have a nut on top of the chuck but some don't some have a key bar the you tap in the shank/arbor of the press to remove the check arbor shank.
But the bottom line is you need to remove the chuck and with some 600 grit or 800 grit paper clean both parts, once you have the parts clean, NO marks inside or out, move the table to one side and get a hard wood block and some WD40 and drive the chuck back on, it will take a sharp hard blow to set it in place.
Besure to have the chuck open to the max DON'T hit the chuck with any metal tools top or bottom of the chuck, it will damage it and it will become junk.
You must take care when you work on the chuck it's like a big watch and made the same way.
If you need to tap on the chuck use some hardwood and cut it so it fits the main part of the chuck NOT the outside ring just the main body.
It also needs a sharp blow to get it off
Also don't over clamp it in the vise use your wood vise if you have one or some wooden blocks for this job,they are made of high end tool steel and will snap like a walnut nut...
Just a small note**** this is quick way to check the shaft out,if you a maget around take a metal rod put some elec.tape on one end then place the maget with the rod on it, place it on the drill press so it can just touch the shaft, turn the pulley by hand and view the end on the rod, should NOT be a gap at any point, if so take it back to Sears for a replacement.
Hope this helps