Yes you can do that or there are static site generators backed by MySQL too. So your data and configuration site can still be dynamic but your served site will be completely static.
The only difference between this and adding a cache is that the cache is another piece of software in your production stack.
For WordPress? No. I don't use it. I just know SSGs for it exists and not enough people/companies use it (when they're using WP in the first place). :)
When my friends webserver died and they had no backups, I found the wayback machine was s good (historic) static site generator ;-) Just mirror it from there and voila.
The only difference between this and adding a cache is that the cache is another piece of software in your production stack.