How to create preconditions by Stream/Component
Accepted answer
You mentioned that you need all team members to be able to deliver to all streams. If you grant users permission to deliver change sets in your project, then they will inherit those permissions in any team areas within the project. So it sounds like you'll be able to achieve your goals.
Comments
What if I have more streams than project areas? I think this solution doesn't scale. It would be better in my opinion if the precondition "Require Work Items to Match Query" allows you to configure not only the Component, but also the Stream
Note that you don't need additional project areas ... just additional team areas (still overhead, but not as heavyweight as a new project area). But requesting the built-in precondition to allow you to configure per stream would be a reasonable enhancement request (see Sumant's comment if you need a link for that).
2 other answers
Comments
I don't think it is possible to set different preconditions for different teams of the same project area. Even when that's possible, I need all streams to be owned by the project area.
Here's the link to raise a RFE (Request For Enhancement):
It is possible to set different preconditions for each team area in the project area. Just for interest's sake, why do you need all streams to be owned by the project area?
Because all teams should be able to deliver changes to any streams.
Did you follow Jared's guidance on clicking the "Customize the process" button?