Columbia model 105 LP player

[bob91343]

From the picture of the cartridge it appears that a stylus will go in from either side, so that might mean it was a flipover type. Not sure, really.

The compromise 2 mil styli tended to tear up the records. It played either groove size equally badly but I think was most destructive with LPs.

Yes the spindle diameter is somewhat critical, although most of those players ran the records a bit faster than 78. So shaving down the spindle might be a solution. It does, as you found out, take precision tools and a steady hand to do the job. Chucking the spindle in a drill motor and bringing some emery or sandpaper gently against it might work but that old rubber tends to flake off.