Great catch — this is where the "outside sources" part is crucial. The magnetic field does exist in those regions, but it's generated by currents elsewhere, not locally.

The scalar potential works in regions with ∇ × B = 0 (no local currents). The field itself comes from distant sources — like currents in a wire or atomic currents in a magnet — and just passes through the current-free region where you're calculating.

So you're using the scalar potential as a mathematical tool in the "quiet zone" between or around the actual current sources. The field is real, just not sourced there.