One of my favourite bits of advice on writing I've heard was to do exactly this, keep saved all the portions you cut from your work -- not, the advice said, because you'll ever need to go and recover them -- it's very unlikely that you ever will -- but because if you get rid of them completely you may become convinced that you had written that bit SO WELL and WHY did you cut it and IF ONLY you could remember it!! If you're able to go back and look at them, you can remind yourself "oh yeah, this wasn't that great, there was a reason I pulled this out."
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Re: the actual question,
I think it's a tough thing, because sometimes yeah, absolutely, cut the thing that's not working even if you loved it. But sometimes I find that if the choice is between making a stronger story versus saying the thing I wanted to say, that making a theoretically stronger story isn't worth cutting out the bit that was important to me? Like, congrats, the story is stronger, but it's also not Your Story. Sometimes it's worth trying to figure out how to make the weird piece work.
I've jokingly declared that my whole comic is fanservice, it's just only fanservice for me -- but honestly that's pretty true. If something isn't working (for me) then it needs to be cut, changed, reworked -- I have entire character motivations that have been rewritten, and I try not to be too attached to stuff that I've just always imagined that way before. But I've also had times where I tried to rewrite a scene to remove the silly self-indulgent stuff and discovered..... I'd removed what I was passionate about in that scene. When that happened, I decided it'd be worth trying to get the scenes I liked to work, because without them I just wasn't invested in that piece of the story anymore.
I dunno! I definitely believe in killing your darlings being an important concept, and it's good to have that tool -- to be able to ask yourself "now hang on, is this scene I really like necessary? Am I forcing it in when it's making my story not work?" You can cut those elements, even if you like them, and still make the story you wanted to make -- in fact the story will work better without them!! But then I also think that an element being unnecessary sometimes doesn't mean it needs to be cut. Sometimes you can cut some stuff to make a cleaner story, but it's worth it to make a weirder one that's true to your heart. Finding that line is tricky.
