In search of a component or device that switches multiple lines based on input selection
I recommend that you use three 4066 analog switches, with the 4 inputs of each chip wired in common with the inputs of the other chips, and the four control lines of each chip tied together and connected to your rotary switch.



Although this requires three chips, they are in a 14-pin DIP package so they are easy to wire up, and each can be selected by a single input (no decode), lending itself to the rotary switch in your diagram.

I have used these same devices (in a SMD package) to switch the four active leads (VCC detect, PGEC, PGED and MCLR) of a PIC programmer, and it works just fine.

The chip has a typical Ron (resistance between the inputs and outputs) of 80 Ω, but that does not seem to affect the operation of the programmer.  There are some quad bilateral switches available with a lower Ron, but they are only available in SMT packages.