First concrete, legitimate reason I've seen all thread. Is there something fundamental about jQuery's design that forces it to use eval(), or is it possible a new version could avoids that?

Performing evals is a core part of the design. Take a look at the additional notes section on this documentation of the .html method.

http://api.jquery.com/html/

>By design, any jQuery constructor or method that accepts an HTML string — jQuery(), .append(), .after(), etc. — can potentially execute code

This makes jQuery less safe in general, but I think the CSP restrictions this puts in place are probably a greater issue for most people. I’ve been involved in retro-fitting strict CSPs into a few projects, and more than once I’ve been thwarted by very minor use of jQuery in a couple of places.

It's part of the design of the creation of new elements. But that's just one part of jQuery, and a part I would be completely okay with jQuery getting rid of.