How about making it a “Cut” feature via a modifier key, so it still selects the tile as expected, but also removed the tile. It’d help do somewhat of what he wants and keep it intuitive, as you’d still be doing some kind of brush action. It could display some scissors when holding the modifier key.
Another option (which would work alongside the above), is to let the delete key delete whatever is being selected with the right mouse button. So you hold right mouse button on a tile, press delete, it deletes it. You drag and make a selection with right click and keep it held and press delete and it deletes what’s in the drag area. This would conflict with the current overall “selection”, but you could just have that selection hidden while holding the right button to indicate this.
A third option: allow selecting “no tile” with the brush and have it work as an eraser. This would probably works best as a quick option, like a checkbox in the edit menu.