Why not put CSS styles in body of HTML?

Question

I have seen this topic and have read, for HTML5, it is not good to put CSS styles inside body tag (embedded; not in-line). I know it may cause FOUC, and that it won't be "valid" HTML.

I have big limitations working in a CMS in the ck editor, and I have very little choice but to embed a CSS style tag and rules or spend days inlining. The styles should be minimal, but still it is ridiculous. I am overriding already styled content.

None of the pages will validate whether I do this or not, and the flash is a lesser problem than what currently exists.

The question: Can someone explain exactly (besides the FOUC and not being valid) why is wrong to do this? Is it a problem for SEO?