URL and DOI fields exporting into to Bibliography (?)
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I have offered to help provide that or those styles if someone goes through the trouble of actually checking with some care how they differ from existing styles.
Fixing citation styles according to relevant requirements is the way to fix these issues. Because everyone involved in the Zotero and the CSL project believes this, no one is going to go through the trouble and code a front-end that allows selective de-activation of item fields.
When we label a style "Chicago Manual of Style" we follow what's in that style manual as closely as possible. The style manual explicitly shows examples of articles from JSTOR with doi. While you and many journal editors may disagree, the fact that these are digitized versions matters to the editors of the CMoS. They are not the only ones, btw. - MLA also distinguishes sources by how they are accessed.