  This short note aims at complementing the results of the recent work
arXiv:2302.05396, where Jahnel and L\"uchtrath investigate the question of
existence of a subcritical percolation phase for the annulus-crossing
probabilities in a large class of continuum percolation models, namely the
classical or generalized weight-dependent random connection models. Their work
relates the absence of a subcritical phase to the occurrence of long edges in
the graph, through a somewhat indirect criterium, that stays inconclusive in
some models of interest. We provide a more direct and arguably simpler
criterium, that is always conclusive when considering classical
weight-dependent random connection models.
