For example, your site certainly won't break from missing the 'lang' attribute.
Your site isn't "broken", per se, but it's not as high quality as it could be.
Update: from elsewhere on this thread, Google ignores <html lang>:
> But the language attribute within the HTML markup is something we don't use at all. We've found that this language markup is something that is almost always wrong. So we tend to ignore that.
https://www.seroundtable.com/google-ignores-the-html-lang-at...
FWIW, Chrome's translate feature does care about lang=. Or at least it did last time I checked.
It's one of the things that has no downside, and a small upside. If you leave it out, your site isn't horribly broken. People will always get the content, but tools that process the content might be wrong.