If your whole website doesn't use JavaScript and every request reloads the page, that reloaded page can contain current session id, right? Why do we need to rely on automation which does this thing, with known drawbacks, but can't do that ourselves?
Right, through cookies - unless you want to embed the session ID in every single link or button in the whole page. The way pages "contain" session data like that _is_ cookies! snapetom mentions embedding in query parameters, which while a possible solution, seems even worse to me, as it means sharing a link to the current page you're on leaks your session token. I'm really not sure I'm following what you're saying here.