Why valuations when defining FOL? Why does one need valuations in order to define the semantics of first-order logic? Why not just define it for sentences and also define formula substitutions (in he expected way). That should be enough:

$$M \models \forall x. \phi \iff \text{for all }d\in \mathrm{dom}(M),\ M \models \phi[x\mapsto d]$$

instead of

$$M,v \models \forall x. \phi \iff \text{for all }d\in \mathrm{dom}(M),\ M, v[x\mapsto d] \models \phi$$
It is perfectly possible to define satisfaction using just sentences as you suggest, and in fact, it used to be the standard approach for quite some time.

The drawback of this method is that it requires to mix semantic objects into syntax: in order to make an inductive definition of satisfaction of sentences in a model $M$, it is not sufficient to define it for sentences of the original language of $M$. You need to first expand the language with individual constants for all elements of the domain of $M$, and then you can define satisfaction for sentences in the expanded language. This is, I believe, the main reason why this approach went into disuse; if you use valuations, you can maintain a clear conceptual distinction between syntactic formulas of the original language and semantic entities that are used to model them.