I don't understand where is the problem with crossing voices.
I think this is very important in vocal composition.
There is a book by Percy C. Buck, Unfigured Harmony, which is very old (1911 I think). That is to say: it talks about classic harmony, nothing about impressionism or other modern systems. It's a book I like. It says:
"Never hesitate to cross parts. In vocal writing, pure and simple, there is not much reason for doing so, hence the habit has come to be considered unadvisable. But in writing for strings, or other instruments of large compass, it is most desirable that the different qualities possessed by different ranges should have fair play".
In fact, in instrumental music there are cross voices we can identify easily. Pachelbel's original canon with three violins is constantly crossing parts.
I gess many times crossing voices will not provide a good result. But making it a universal rule... Not for me, sure.
About the quartet... It's nice, but this kind of sound (somewhat modern) is a bit odd in this classic style. This is something I don't see as a problem, you can turn it upside down and make it a novelty. It works better (to my taste) in the quiet movements (Andante, Menuetto).