Submitting a crude working paper in a Ph.D. application
It's been a few years since I've reviewed applications for math grad school, but I don't remember ever looking at "mathematical writing samples" of candidates. (I'm not even sure if this is an option at our school.)  As discussed elsewhere on this site (e.g., https://academia.stackexchange.com/a/67732/19607), actual research outcomes are not typically a major determining factor for admission (though a great letter from someone based on your research experience may be).  Thus, I don't think it will make too much of a difference, provided the rest of your application is solid.

That said, if you do decide you want to upload a writing sample (it could make some impression), my advice is don't upload something unpolished.  That almost surely will not make a good impression.  However, you could write and submit a summary/expose of your paper, which will take less time to write.  If in this case, you should try to have one of your professors look over it if possible.