I quite agree with you that revenge is fine, and also that where the villain's place in the story is to represent the vicissitudes of life their final defeat stands for the audience's potential ability to overcome their own challenges, so if you want an upbeat, optimistic story, you have to have the defeat, whether it would be likely in real life or not.
How do we know the villain isn't just lying
I'm not sure about this one. I mean, I can see how that would work, but I feel decent characterisation should get across the villain's motives. Having to put characters through the wringer to prove they aren't lying is a bit clumsy. Though I guess it's easier to make the villain's change of heart clear in some POVs than others.
seeing the villain suffer makes us sympathize for him
Definitely, and I don't think I'm above using that one myself, however cheap and easy. I'm not sure, though, that this is quite what happens in a redemption arc. It's certainly what happens when the writer wants to reposition the villain as a good guy and wants to get the audience's sympathy, but it doesn't have that element of let's all gleefully watch the suffering because they deserve it. It's more, let's sympathise with this tragic backstory and then excuse all these subsequent failings.
And once he's paid his pittance, he can be considered truly redeemed
This is the one I don't really understand. For a start, you don't get to pay off your karmic debt at the time and place of your choosing, so that you have a clear credit history for your new job with the good guys. Or if you do, it's through efforts of your own, not because fate (or authorial fiat) helpfully supplies the right amount of suffering at the right time.
But the truth is, it may be a difference in attitude. I don't find someone getting away with something unsatisfying, and faced with the authorial imposition of suffering, I generally feel, as in my reply above, if the only way to be redeemed is to suffer, why bother with the good works? But then I tend to like my villains to be villainous under the impression they were doing the right thing, so perhaps the suggestion that anyone who thought they were in the right but changes their mind deserves to suffer makes me uncomfortable - we none of us can be sure we won't change our minds about what was right.
FWIW, though, in your story it seems quite reasonable that the hero's friends don't welcome her with open arms. Actions have consequences, and I certainly don't feel writers should gloss over them. I guess from my point of view the question would be whether you're showing the character of the friends, that this is how they would react, and her character in how she responds, or just putting on a circus display where the audience gets to applaud their stand-ins being cruel to someone who 'deserves it'.
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How do we know the villain isn't just lying
I'm not sure about this one. I mean, I can see how that would work, but I feel decent characterisation should get across the villain's motives. Having to put characters through the wringer to prove they aren't lying is a bit clumsy. Though I guess it's easier to make the villain's change of heart clear in some POVs than others.
seeing the villain suffer makes us sympathize for him
Definitely, and I don't think I'm above using that one myself, however cheap and easy. I'm not sure, though, that this is quite what happens in a redemption arc. It's certainly what happens when the writer wants to reposition the villain as a good guy and wants to get the audience's sympathy, but it doesn't have that element of let's all gleefully watch the suffering because they deserve it. It's more, let's sympathise with this tragic backstory and then excuse all these subsequent failings.
And once he's paid his pittance, he can be considered truly redeemed
This is the one I don't really understand. For a start, you don't get to pay off your karmic debt at the time and place of your choosing, so that you have a clear credit history for your new job with the good guys. Or if you do, it's through efforts of your own, not because fate (or authorial fiat) helpfully supplies the right amount of suffering at the right time.
But the truth is, it may be a difference in attitude. I don't find someone getting away with something unsatisfying, and faced with the authorial imposition of suffering, I generally feel, as in my reply above, if the only way to be redeemed is to suffer, why bother with the good works? But then I tend to like my villains to be villainous under the impression they were doing the right thing, so perhaps the suggestion that anyone who thought they were in the right but changes their mind deserves to suffer makes me uncomfortable - we none of us can be sure we won't change our minds about what was right.
FWIW, though, in your story it seems quite reasonable that the hero's friends don't welcome her with open arms. Actions have consequences, and I certainly don't feel writers should gloss over them. I guess from my point of view the question would be whether you're showing the character of the friends, that this is how they would react, and her character in how she responds, or just putting on a circus display where the audience gets to applaud their stand-ins being cruel to someone who 'deserves it'.