I don't think you could take over the company with a jira token. Another factor for consideration with turnstiles is disability access and fire egress. Those are covered by building code but since this is a parable, it's worth noting that physical security has often caused tragic stampedes that have killed many.
You are right, it's much harder to compromise a system with the jira token, which is why it was the solution for the username/password stored as cookies. Plus the token was never exposed to the client.