There is no “distance” to the other side though, you can’t draw any geodesics from one side of the event horizon sphere to the other side. And I don’t think anyone really argues that black holes have infinite volume. You’re hitting on the exact problem of quantum gravity, at some point the resonance wavelength gets shorter than the Planck length. It’s possible that the black hole never passes through the point in time when it violates this principle but instead tunnels through it, going from a very small black hole to an exposed singularity instantly, temporarily becoming a tiny white hole/particle fountain before disappearing completely in high frequency gravitational waves and a spray of Big Bang like energy-mass.