HTML5/CSS3 classless framework

It sounds like what you want is not a CSS framework, but just a simple custom stylesheet.

Not to discourage your effort in looking for something that already exists, but you're better off writing it yourself. Perhaps start off with one of the many existing reset.css. Default styles without the ability to hook in (or out) via class names are not very flexible.

You really shouldn't avoid classes, you should take advantage of them. No framework is going to know where and how your HTML elements are nested or what they're supposed to look like based on tag name alone, that's one reason why we usually use classes.

If you pay close attention to the semantics of the new HTML5 elements, you'll see that header doesn't always mean "the top of your page with your logo", footer is not just an element for the bottom of your whole page, and aside doesn't always equate to "sidebar".

I found a very similar need and ended up making my own project: https://github.com/oxalorg/sakura

You just need to drop in sakura.css and you're good to go, no changes required to HTML.

Here's a demo: https://oxal.org/projects/sakura/demo/