Is there a known obstruction to the prime generating function being modular?

Let

$$ P(\tau)=\sum_{p\ \mathrm{prime}} q^p,\qquad q=e^{2\pi i\tau}. $$

The coefficients of $P(\tau)^2$ count ordered Goldbach representations:

$$ P(\tau)^2=\sum_{n\ge0} r_G(n)q^n, $$

where $r_G(n)$ is the number of ordered pairs of primes $(p_1,p_2)$ such that

$$ p_1+p_2=n. $$

This is formally analogous to the classical theta function

$$ \Theta(\tau)=\sum_{m\in\mathbb Z} q^{m^2}, $$

whose fourth power has coefficients counting representations of integers as sums of four squares.

My questions are:

1. Is there a known theorem showing that $P(\tau)$ cannot be a classical modular form?

2. More generally, can $P(\tau)$ (or the von Mangoldt generating function $$ L(\tau)=\sum_{n\ge1}\Lambda(n)q^n $$ ) be realized as a quasimodular form, mock modular form, harmonic Maass form, automorphic distribution, or some other automorphic object?

3. Is there a conceptual obstruction explaining why the theta-function approach for the four-square theorem cannot be adapted to Goldbach?

Any references would be greatly appreciated.