It's very much a case-by-case kind of thing. My channel F has a busted 2 button, but otherwise works fine. And I might replace it, becuase it's all but impossible to find someone who can repair such an old & obscure system!
On the flipside, despite it being more expensive than a replacement, I had my original Xbox repaired when it started turning itself on & off. The unit I have was the one my fiancé bought new back in the day, which then passed through all his siblings before ending up with me. There's a lot of sentimental value in keeping that specific machine running, over the cheaper option of a new deck. Plus when I picked it up, the clerk commented that it was a 'surprisingly well maintained Xbox', which was nice.
I will say, of the machines that I have gotten rid of over the years, only one was actually thrown out- that would be the NES from my sister's friend that was painted black with acrylic and had a replacement power supply running the wrong current. There was nothing to salvage there. Anything else I've put into the hands of another collector or a shop, that might make use of it. If I can't keep it running, perhaps they can- or at least get a different machine going using salvaged parts.