Can a propositional threshold connective be expressed by standard connectives?
The idea is to use the following steps:


Construct an NC1 circuit computing the binary representation of $x_1+\cdots+x_n$, see for example these lecture notes (Theorem 7). This circuit implements carry-save addition.
Convert the NC1 circuit to a polynomial size formula, see for example these lecture notes (Proposition 1).
For each $k$, it is now easy to construct a polynomial size formula that tests whether $x_1+\cdots+x_n=k$.
Use up to $n$ of the formulas from the previous step to come up with your desired formula.


You can in fact implement the last two steps more efficiently, using $\log n$ rather $n$ copies of the preceding formula, by using a comparison gadget; details left to you.