The off side of the switch probably makes continuity until pressing off opens the circuit.
This is just a typical cheap phosphor bronze contact, likely at or above good design rating for current.
When these contacts get warm, they oxidize, then the surface doesn't conduct as well, so it gets hotter and oxidizes ever faster, etc.
This is why the off switch was so warm. There was high resistance at the switch.
The usual way these die is to get so hot the plastic surrounding melts, no longer holding the fixed contact where it needs to be.
If you like the lamp buy a better switch, adapt a better switch, or buy a stock of them for the future.
Beware of the extravagance of cheapness in new lamp purchasing. This is what happens.
How old was this lamp/switch?