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The nature of the superfluid-to-Bose-glass (SF-BG) quantum phase transition, occurring in systems of interacting bosons immersed in a disordered environment, remains elusive. One fundamental open question is whether or not the transition obeys conventional scaling at quantum critical points (QCPs): this scaling would lock the value of the crossover exponent $ϕ$ -- dictating the vanishing of the superfluid critical temperature upon approaching the QCP -- to the value of quantum critical exponents for the ground-state transition. Yet such a relation between exponents has been called into question by several numerical as well as experimental results on the SF-BG transition. Here we revisit this issue in the case of the $S=1/2$ Heisenberg antiferromagnet on a site-diluted cubic lattice, which lends itself to efficient quantum Monte Carlo simulations. Our results show that the model exhibits a percolation transition in zero applied field, with the correlation length exponent $ν= 0.87(8)$ and $ϕ= 1.1(1)$ consistent with 3d percolation. When applying a sufficiently strong magnetic field, the dilution-induced transition decouples from geometric percolation, and it becomes a SF-BG transition; nonetheless, the $ν$ and $ϕ$ exponents maintain values consistent with those of the percolation transition. These results contradict the conventional scaling, which predicts $ϕ\geqslant 2$; and they suggest a close connection between the SF-BG transition and percolation.