That wouldn't solve today's traffic spike, and there are probably some lower-hanging fruits that should at the very least keep the site online. Anything sold in the last five years with an ethernet port should easily be able to handle a static site even during rare spikes in traffic.

But the Cloudflare DNS configuration isn't something to be scared of: I used to feel similarly nervous when fiddling around with DNS, but haven't had a single problem configuring 20+ sites with Cloudflare. (I would, however, be slightly more nervous about google's reaction to the move to a subdomain this would require.)