The argument is made here at the start of section 4, on types. From what I understand, he didn't really show that delta was finitely satisfiable because delta must hold for all v, not just a specifically selected v. Here, he treats v like a constant symbol. Is there something I'm fundamentally misunderstanding?
When working with types in model theory, we consider the free variables in that type as constants. A realisation for that type is then an interpretation for these constants.
So being finitely satisfiable means that for any finite subset of the type we can find some realisation of its free variables (which you should this view as constants), which is exactly what is done here.