Quill and spindle.....
The spindle shaft is rotated by the large bevel gear. A hardened key, fitted inside the gear body, transmits force via the keyway in the spindle shaft and allows the
spindle to be raised or lowered.
I see in your picture a key on the spindle shaft. Is that the drive key, broken and out of position?
The body of the gear rotates in a babbit bearing. It uses a phenolic thrust bearing which is pinned to prevent it's rotation.
Wear of the babbit or thrust will cause the gears to run out of alignment. Gear alignment is critical to prevent wear.
The keyway in the spindle and the drive key need to be checked for wear and dirt as well as the bearing areas.
The spindle shaft is somewhat hard and the ID of the cast iron quill acts as a radial bearing so the clearance should be tight.
Perhaps 0.0015" .
There's ball thrust bearings top and bottom of the quill but if the adjustment nuts were not correctly set they could be damaged or have
wobbled in the seats wearing the ends of the quill. Dirt in any area of the quill assembly will have caused wear problems.
I don't think the thrust bearings carry any radial thrust.
The quill OD should have a sliding fit in the cast housing of the drill press. That housing is split and has a adjustment bolt to snug the fit up
but the OD of the quill and the ID of the housing are worn too much then the quill can have too much horizontal travel.
That's a general description so you will need to disassemble, clean, and measure everything to know what repairs are needed.
My drill press is still disassembled so its no problem if you would like pictures of the components.
Jim