oh, another "design over SEO" debate. groundbreaking.
I'll ask the same thing - why do you think it'll tank rankings? Not being difficult, genuinely curious.
How do you know it will affect rankings and why?
you've already decided it's a bad idea, but you're asking how to convince him. that's backwards. first, figure out if it actually matters. removing a footer and top nav doesn't automatically nuke your rankings. Google doesn't care about your site's aesthetic - it cares about crawlability, content, and user signals. If your core pages still have internal links, your navigation still works, and you're not hiding content behind JavaScript hell, then stripping some chrome won't move the needle.
the real question: is the site still usable? can users find key pages? can Googlebot crawl them? If yes, then your boss's "physical appearance" fixation might actually be fine. If not, sure, push back - but with actual data, not vague assumptions.
Stop playing the "Google won't understand" card. It doesn't work like that