[OJFord]

I'm not suggesting a pairing, but I think expecting that it might be 'on' is perhaps a bit, er, off.

To me (BrE), 'gone off' is like 'has developed off-flavours', which is like off-course & on-course but also off-side & near-side.

[pessimizer]

"On-side" is definitely a thing people say.

"Off" can also mean "away", though. If you "go off", you can "come back."

[OJFord]

Oh yes that's true - but that's antonym to a different 'off-side'!