A
JSF page is not bound to a single backing bean. If it was, there would have been no need to supply the bean ID as part of the EL in value references. You can reference as many different backing beans in a JSF View as you like, and contra-wise, a backing bean may be referenced by multiple JSF views. It's up to you to co-ordinate the communications between the beans, however.
It's entirely permissible to create a logic-only backing bean. I have one or two of those myself. They supply methods that don't need any stateful backing data.
The secret of how to be miserable is to constantly expect things are going to happen the way that they are "supposed" to happen.
You can have faith, which carries the understanding that you may be disappointed. Then there's being a willfully-blind idiot, which virtually guarantees it.