Why shouldn't I use a JSF SessionScoped bean for logic?

I'm developing a java EE web app using JSF with a shopping cart style process, so I want to collect user input over a number of pages and then do something with it.

I was thinking to use an EJB 3 stateful session bean for this, but my research leads me to believe that a SFSB is not tied to a client's http session, so I would have to manually keep track of it via an httpSession, some side questions here . . .

1) Why is it called a session bean, as far as I can see it has nothing to do with a session, I could achieve the same by storing a pojo in a session.

2) What's the point of being able to inject it, if all I'm gonna be injecting' is a new instance of this SFSB then I might as well use a pojo?

So back to the main issue I see written all over that JSF is a presentation technology, so it should not be used for logic, but it seems the perfect option for collecting user input.

I can set a JSF session scoped bean as a managed property of all of my request beans which means it's injected into them, but unlike a SFSB the JSF managed session scoped bean is tied to the http session and so the same instance is always injected as long as the http session hasn't been invalidated.

So I have multiple tiers

1st tier) JSF managed request scoped beans that deal with presentation, 1 per page.
2nd tier) A JSF managed session scoped bean that has values set in it by the request beans.
3rd tier) A stateless session EJB who executes logic on the data in the JSF session scoped bean.

Why is this so bad?

Alternative option is to use a SFSB but then I have to inject it in my initial request bean and then store it in the http session and grab it back in each subsequent request bean - just seems messy.

Or I could just store everything in the session but this isn't ideal since it involves the use of literal keys and casting . etc .. etc which is error prone. . . and messy!

Any thoughts appreciated I feel like I'm fighting this technology rather than working with it.

Thanks