I started this before you got your chuck off and spindle apart and just got back to it now.
If your lathe has been taken apart at some point in the past perhaps some of the end float may be caused by the 9-32 threaded end collar not being tight enough or if it is tight one or more of the parts on the spindle may not be shouldering up in their proper place.
Perhaps a space is missing or in the wrong position etc.
While there may be some wear in the bearings I would suspect that if the other parts that go on the spindle the end collar could be tightened up to eliminate the float.
You don't want it so too tight to damage the bearings just enough to eliminate the end float.
You might want to check if you can get the current bearings in place and test the run out and end play and be sure there are no other issues before ordering new bearings.
Those bearings may be not so bad if they are adjusted properly .
I think there is a user manual on archive.org or the Hathi Trust online library for those machines that may tell how to set the preload on the bearings .
I can't look for it right now.
Someone else may know and post about that later.
Vintage machinery has several parts books for Craftsman machines.
This appears to be the one for yours and shows how it should be put together.
also a catalogue picture on page 28
With regards to belts you may want to consider Gates or other industrial belt suppliers .
If I remember you want to use the 4L size on these,. since they are strong enough for that size if machine and are more flexible taking less power to drive around the smallest diameter pulleys and make changing steps easier.
The width of the pulley grooves will dictate the belt width.
Someone may correct me on this since it is a long time since I had anything to do with these machines.
P.S. I didn't see Joe Michaels' post before I posted this he has expressed my thoughts more clearly than I did.
I found the manual I was looking for and it tells about setting up the Timken bearings here
How to use a lathe. (Dark line in middle from tape used to protect scanner from scratches from metal wire) No Copyright renewal found in Stanford or LOC online...
archive.org
Jim