Well, I think the main difficulty with SSR is you'd think it would be more of an incremental step. I don't think it's so much a "separation of concerns" issue (i.e. just having some of your presentation logic running on-server doesn't mean it's necessarily not isolated logic) as it is a "wow I have to rewrite significant parts of my application to take advantage of this" issue. Some frameworks definitely do a better job with this than others IMO, but not everybody wants to (or should) migrate their React app to NextJS (for example) just to get SSR.
You just described exactly why most web devs strongly prefer to build SPAs today instead of using SSR (separation of concerns).