It does break search. A traditional SPA returns to the client an empty HTML page (except for header stuff) and a script to parse. The script then populates the DOM with the elements, problem is that crawlers are not really good at waiting for the "render" to happen, and running JavaScript just to see what's in the page opens a whole other can of worms. SPAs are awful for search engines and that's why we get the hybrid monsters that Next.JS, Nuxt, SvelteKit and co. produces nowadays (not saying they are bad from a user perspective though, much prefer web apps made with them over client-side rendered SPAs).
We spent the last decade trying to fix the whole mess that client-side "rendering" created in the web, we'll probably spent the next decade trying (and failing) to standardize things such as web components, declarative shadow DOM, among other things. That's because we as developers keep insisting in complexity: hydration, SPAs, bundlers (and in consequence bundle bundlers such as Vite). The only true solution is to bet in simplicity, but that's not sexy, and sex sells.
In conclusion: The React revolution and it's consequences have been a disaster for the human race.