You're using a specific example of a billiard ball scenario

The billiard ball metaphor/analogy is not a scenario. It is closer to the conception/model of classical deterministic physics. Everything is a billiard ball, including forces.

Like i said, please try to focus on the topic at hand. There is no benefit in discussing this tangent for now - it's just distraction. It is well worth circling back to though.

Is an object being pulled by another experiencing a pushing force?

I'm happy to circle back to this topic once we are finished discussing the first one. Are you saying you have nothing more to say / question regarding weight being an intrinsic and inexorable property of matter and now understand what i'm saying?

Assuming that is the case, then the answer to your question is perhaps best answered socratically/dialectically :

Can an object be pulled by another object if the leading object itself is not pushed? (think mechanically for now)