Unless you’re talking about SSG and not SSR, I think one potential issue would be that React’s SSR process is pretty slow and costly, at least from what I understand. I have to confess I haven’t heard of a pure-SSR React app before, as you wouldn’t get many of React’s benefits if it’s just being used as a templating language.
It's really no more costly than Django/PHP/Ruby mustache-style template rendering. I consistently see 100-150ms last byte timings from a Next.js app I host on Vercel.