Well, it probably could. But in this Czech sentence "to" is the subject. Therefore the best translation is "it": "(To) neskončilo = It didn't end"
Why we can´t use present perfect? -> It has not ended with one cup? It´s still unfished..
While I don't think your suggested translation is grammatically incorrect -- and neither did a handful of online grammar checkers -- using "finish" rather than "end" sounds very odd.
Please report a complete sentence. I would consider it even a matter of etiquette, I also don't just dismiss your utterance with two words.
While I prefer the given translation, could it also be translated as "It was not finished with one cup"?
I am not a native Czech speaker, but here's my take on this. While the meanings of the two sentences would be about the same in English, your suggestion is in the passive voice, unlike the Czech sentence, and it is not an accepted alternative at this time. But I will add it if the CZ natives on the team feel it is appropriate.