What's so awful about switches? They work predictably, they are extremely clean, and you can pull off the complicated attenuation and gain control needed for that circuit very simply. It's hard to say that about pots. I found some dual deck 11 position Grayhill rotary switches at Mouser that were less that $20, but were sealed, with gold over silver contacts - essentially ideal for decades of use. I bet other places have less fancy switches for less money.
How cheap does it have to be?