Scratched my head quite hard thinking what you said here, made some rough sketch, something looks or similar like this Rufus ?
I don't think you quite have it. They call it a U/I converter but it is easier to think of it as a simple op-amp buffer as below.
The op-amp forces point X to the same potential as the +ve input, the +ve input is earthed so point X is forced to be earth also. Point X is called a virtual earth. If you push current in or out of point X the op-amp will compensate and keep it at earth.
If you add resistance in the op-amp feedback path it works the same but the op-amp output voltage has to change (a lot) more. The current in the feedback path has to match the current pulled or pushed from the virtual earth. The voltage across a resistor in the feedback path will be proportional to that current.
As long as you measure the voltage across a stable part of the resistance (Rref) it doesn't matter if there is additional unstable resistance (Rswitch) in the path.
Oh, and if the meter has high input impedance it doesn't matter if there is a small unstable resistance in series with it. One half of the 4052 feeds the op-amp output to the required resistor, the other half feeds the voltage from the required resistor to the meter.