But what about FastCGI? Let's say my web app is self-contained binary and I'd like to deploy it to a web server. I'd like to keep the binary running rather than use the start/stop approach of plain CGI. Is FastCGI a good alternative? Or it too complicated? The application server approach also feels like overkill. Or is it?

There seems to be surprisingly little info on FastCGI. The official website doesn't even exist any more, but you can find it via archive.org or via GitHub [1].

If you've used FastCGI using Apache, NGINX etc how did you find the experience? When would you recommend it?

[1] https://fastcgi-archives.github.io/