why? would be easier to use apache or something else I would think. The built in jekyll server is not really a production server.
when you close the console does jekyll stop running? not sure how PM2 would help, other than to start it when the server restarts, though I am no PM2 expert that is for sure.
A dynamic website would need a running server. A STATIC site generator does not. Please try to grasp the concept of SSGs, as I think you are missing the essence.