IMO the pivot is very underutilized by Falcon players in general. It should be used in the obvious cases such as fthrow -> pivot -> reverse uair and doing pivot fsmashes to counter people jumping up from hanging on a ledge but it can also help with spacing a lot more than people give it credit for. I've been incorporating it into my game a lot more and it does wonders for being able to get out of the way of attacks and then follow up with a grab.
show me a good falcon that doesn't pivot.
I use it in the examples you posted + tons more all the time.
I like to fthrow -> pivot bair at low percents against heavies.
against noobs who use slow smashes to edgeguard i like to roll in from ledge -> pivot fsmash
gentleman -> pivot grab is **** at high damages. or just the first two jabs. Yea now that i think about it the gentleman has set kb (rite?) and prolly wouldn't combo into pivot grab. But you can still do it. ****
fthrow -> double pivot regrab, or, fthrow -> pivot -> dash grab on heavies at 0%
I like to pivot off stage to, either to edgehog or to pivot bair or something.
edit: apparently **** is banned but tit is not.