I have this problem all the time. I'm changing it though. I do understand the feeling of "this isn't worth writing", and I also see the point others are making in that our time is too valuable to waste.
However, I am trying to change that mindset. I am finishing everything I start now. I find that when I tell myself "this isn't worth writing", it's very rarely an actual case of "worthless" writing. How do I know? Because I've noticed I keep coming back to the same pieces of writing I never finished, starting them in a different way, long after I decided not to continue the piece.
I think it is best to really try to finish everything you start. I think a lot of our own writing that we give up on, we give up on because it is hard to keep creative juices flowing (like when you get started and you feel like what you are writing is almost alive, it is that vivid, but then you get to that inevitable low point where everything starts to peter out) or we are afraid of not being very good at writing, afraid of hurting loved ones, etc. Basically, there is another reason we'd rather not continue a piece, and I'm guessing those problems won't be solved until you actually resolve to finish what you started.
I suppose that's not the case for everyone, but that's my opinion on it. I hope you found it valuable in some way.
I couldn't count the number of stories I haven't finished because I got half way and had run out of things to say or because, on closer inspection, I was just doing a generally shit job of writing it.
But I don't think of these as "not worth writing" because if they weren't, I would never have started. It only means that I'm not ready to write them and I need some more time. There are stories I will not just finish but likely start all over again when I am a better writer than I am now. That's just how it is.
But if you think you're doing a bad job or that there's nothing to say with a story, then don't write it. Don't force it on without serious thought and massive revision. If you don't like writing it, nobody will like reading it.
No, usually if I feel a story is not worth writing, it means that I don't want to continue it.
If the story is feeling flat or pointless, but I think it has potential, then I'll fix it up and continue it. I'll make it worth writing.
There's not really much point posting a first chapter of an unfinished story to ask if you should continue it tbh. You should continue it if you want to. If someone says, "no" are you really going to go, "" and if they say "yes" are you really going to continue it if you don't want to?
Why are you trying to avoid clichés? Clichés have become clichés because people enjoy reading them. Do them well, add your own twist to them and you can have a good story. Avoid them completely and stories tend to be weird and/or boring.
As for me I've never had a story I thought not worth writing. I've had stories I don't think are worth editing, worth polishing for some future publication. But every story is worth writing, I suggest writing it all then deciding, maybe give it to a friend to decide.