> I am cleaning the interrupt flag right before RETFIE with BCF
> INTCON, 01h (which should clear INT0IF bit)

It would be better to use the flag name, BCF INTCON,INT0IF
so that you avoid typos, make it easier for yourself to read in the
future and so that it's clear to anyone else

> If this keeps working well, I might just try replacing that big switch
> with something smaller and with some better push-to-'click' feel,
> before doing anything else with extra IC gates

Tact switches can be had with that "click" to them, and they
seem to be better than those that are "soft". And at least they
give the user some feedback. Probably the worst are rotary
and slide switches

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