There are today on Stack Overflow 16,491,952 questions of which 7,739,308 have no accepted answer but do have 8,885,956 answers and have a grand-total of 8,349,785,732 views (== money). A total of 1,341,062 questions without an accepted answer have a score > 1 meaning at least 2 users found that question useful.
If the votes matter, 1,341,062 questions should be deleted by moderators. We currently have 26 moderators which are 100% occupied handling all the other flags being raised. Adding this deletion task to their workload will probably keep them occupied for 6 to 8 years. Alternatively you could argue this deletion should be vetted by community review. In that context I refer you to the close vote queue (currently at 7K but has seen numbers into 150K) that also never gets really cleaned. I don't have high hopes for a delete vote queue.
While I'm all in for a better signal/noise ratio I have a hard time justifying given these plain statistics that simply deleting questions without an accepted answer is useful.
There are already automatic mechanisms in-place to remove questions without an accepted answer. Those are caught by the roomba if those posts aren't voted on.
I personally wouldn't judge an answer (or question) on it being accepted or not. The votes is the only thing that should matter as it is much more likely that plenty of people agree on what is the most useful answer while for an accept vote only a single user thought it was an useful answer. Beyond that accepting an answer isn't mandatory and shouldn't be in my opinion.
tl;dr; No, questions with no accepted answer can stay as they have value. The current rules around the roomba already achieve some reduction of noise. Users can help that process a bit by down voting more (or by not casting pity upvotes)