I wish you success for  https://sqwok.im .  You can let us know when it's ready.

>use-case you mention which is allowing site-owners to add a chat interface for their own pages (c2c).

Couple of sites might prefer c2c engagement, especially if they trust their product and if c2c increases engagement e.g. Gaming, Video streaming etc. But for major sites like E-commerce websites as the OP asks, the publisher wouldn't be willing to integrate a c2c chat and so there we cannot depend upon the site owner for a solution.