I use

    git push

and that deploys my static site, which I think counts as "Heroku but even easier".

For this to work, in the bare repo I push to, over ssh, the post-update hook automatically pulls into a non-bare repo:

    #!/bin/sh
    set -e

    echo -n 'updating... '
    git update-server-info
    echo 'done. going to foo'
    cd /home/bar/public_html/absolute/path/of/a/non/bare/clone
    echo -n 'pulling... '
    env -u GIT_DIR git pull
    echo -n 'updating... '
    env -u GIT_DIR git update-server-info
    echo 'done.'

You can use rsync too, but I think Git is substantially better for this kind of thing.

Now of course this assumes that you already have a server machine somewhere, which you can ssh into, which is running a web server, and which is already in DNS. I've been in that situation for 26 years now, except that at first I used telnet. So I kind of take it for granted.

It sounds like maybe you're looking for a way to not run your own web server process, instead sharing the burden with other people, like a PHP shared host from 15 years ago. If that's what you want, maybe you should share the server with a few friends? Like, two to eight? That way you don't have to give up the ability to solve whatever problems come up with the server.