Why isn’t ‘because’ a logical connective in propositional logic?

It is because 'because' is not truth-functional.

That is, knowing the truth-values of $P$ and $Q$ does not tell you the truth-value of '$P$ because of $Q$'

For example, the two statements 'Grass is green' and 'Snow is white' are both true, but 'Grass is green because snow is white' is an invalid argument, and hence, as a statement as to the validity of that argument, a false statement.

On the other hand,'Grass is green because grass is green' is a true statement as to the validity of this as an argument, but yet again it involves two true statements.

This shows that with $P$ and $Q$ both being true, the statement '$P$ because of $Q$' can either be true or false, and hence it is not truth-functional.

Why isn’t ‘because’ a logical connective in propositional logic?

Is this because the equivalent of ‘because’ is the argument of the form ‘if $p$, then $q$’ ?

Exactly.

Either the connective "because" is truth-functional, in which case it is the same as "if..., then...", or it is not truth-functional, in which case we need a different way of modelling it.

See e.g. Counterfactual Theories of Causation.

See also Arthur Burks, The Logic of Causal Proposition, Mind (1951).

I agree with the other answers, however I want to add that the closest thing might be the turnstyle symbol $\vdash$, although this is usually read as "yields", and thus points the other way. If I write

$$A \vdash B$$ this is read as "A yields B", or "knowing A, I can prove B". If you wanted to encode because, you could probably read it backwards as "B because of A".

Note however that this is not used as part of a logical formula, but as a shorthand between formulas when writing down a proof. So $A \vdash B$ is no longer a formula, but rather a statement on how to prove $B$. (In most of the rest of mathematics, you would write $\Rightarrow$ in your proof instead, however in logic this is of course easily confused with the implication inside formulas)