How to make one switch control multiple identical subcircuits?
What approach would you recommend?

A likely candidate is an analogue switch. You can get multiple independent analogue switches in one IC package and each can be &quot;switched&quot; on or off independently or together. They are also called transmission gates and when several are used together to form a multi-pole switch it is called an analogue multiplexer/demultiplexer.
You can get ICs that contain several (as already mentioned) and you some ICs have normally closed and normally open types in the same package. You can choose the on resistance to be from around 1 ohm to several hundred ohm and the off-state leakage current can be a few nano amps.
Usually, the control lines that are used to switch them on or off are digital voltage referenced making life easy if you want to control it from a microprocessor. Switching speed can be many MHz (if you need it).
One constraint - it has to be capable of running from the same power supplies as your circuit and the analogue voltages that are used have to be within those power rails.