Ah... I thought he meant HTML file attached to the email, not a "HTML email".

I meant HTML files attached to the email. They print badly. They don't display reliably. Use PDF.

Even if you use vanilla HTML and/or test cross-browser display?

Yes. I've seen way too many mangled CVs. Stick to PDFs unless the company in question requests a specific format. Beware of agencies deliberately changing your CV regardless of format (see my other comment).

There are exactly no mainstream browsers that print even fairly basic HTML pages reliably, IME. Typically, the scaling and positioning of page elements ranges from poor to seriously clipped/unreadable once you get beyond the most trivial formatting.