Why is the decay of a neutral pion $\to$ electron-positron loop suppressed?

In the pion reference frame the two outgoing leptons are very boosted, hence helicity and chirality almost coincide. The angular momentum conservation forces them to have opposite spins, since the pion spin is zero. Therefore, they will have the same helicity, which is highly suppressed in this kinematic regime, because of the vector nature of the QED interactions (see for example Thomson, Modern Particle Physics, chapter 6). Just my two cents.

The weak decay is not forbidden. You can have $q\overline q \to ZZ \to e^+e^-$ using a loop like in the EM decay. You need a loop to conserve angular momentum, because a single $Z$ or $\gamma$ has spin $1$ while the $\pi^0$ has spin $0$. If an EM decay is possible, it is so much faster that it will dominate. You can see this in the overall $\pi^0$ decay rate compared to the charged pion decay rate. The neutral pion decays $10^9$ times faster because it is an EM decay. The charged pions have to decay weakly as there is nothing they can decay into electromagnetically.