We provide a relatively compact proof of the BPHZ theorem for regularity structures of decorated trees in the case where the driving noise satisfies a suitable spectral gap property, as in the Gaussian case. This is inspired by the recent work [LOTT21] in the multi-index setting, but our proof relies crucially on a novel version of the reconstruction theorem for a space of "pointed Besov modelled distributions". As a consequence, the analytical core of the proof is quite short and self-contained, which should make it easier to adapt the proof to different contexts (such as the setting of discrete models).