Ask HN: Why do static sites need to be generated on the server?

I'm in the process of building a Hugo site, and I'm at the point where I'm looking for deployment options.

My assumption was that I'd mostly be looking at fancy ways of just pushing my /public folder to a server. I run Hugo commands locally, maybe I push to a git repo, and that triggers a push of my public folder.

Instead, I'm typically being instructed to .gitignore my /public folder, push everything to a server (sometimes via a git repo), and then allow the server to run it's own build command and then deploy?

Can someone help me understand the benefits of building the static site on the server as well? Certainly it can't just be for URLs because that seems like a problem solved easier in other ways.