Sure, old websites will stop working, but just as we shut down our FM radio and analog TV to make room for new amazing technology, and therefore rendered several generations of radio and TV devices useless, the internet will need to learn to live with the fact we can't possibly support its entire spectrum of functionality forever.
However, I see this as a tooling and ergonomics issue. Despite what people say about certbot, it is still not 100% trivial to use it. If your stack differs slightly from the canonical HTTPS server, you're gonna have issues. I'm not saying these problems are inherent to certbot, just that there is room for improvement.
Once we have normalized how certificates are signed and verified, and use encryption methods that can easily be tuned for increased computing power, and more importantly, once HTTP 1.0 is no longer an expectation, I believe the amount of manual work needed to maintain a server will go down again.
We're at a transition phase and we all know these are never easy (except if you're Apple and you're transitioning CPUs architectures, apparently).