While MC-112991 is won't fix, the opposite behavior happens in the offhand:
When holding the item, and not riding a pig/strider respectively...

• In the main hand, it increases the use statistic. (MC-112991)
• In the off hand, nothing happens.
• In both hands, nothing happens.

This behavior should be consistent across hands, and if off hand not doing it is not a case to fix, it should not prevent the main hand at least.

How to reproduce:

• /scoreboard objectives add Use minecraft.used:minecraft.carrot_on_a_stick
• /scoreboard objectives setdisplay sidebar Use
• Use a carrot on a stick in main hand, leaving off hand empty
  -> Score increases (MC-112991)
• Hold one in both hands, and use again.
  -> Score doesn't increase.