@Justme is correct that the signals MAY be affected by adding external connections (it's part of the risk of our not having all the available information to work with), but there is a reasonable chance that a 'high impedance when off' connection would be acceptable. Balance between circuits MAY be important.
Solid state switches may be usable but without ...
SD and eMMC signals are high speed, low voltage signals and messing with them could ruin the signal integrity. It's not possible to switch them with a mechanical or solid state relay.
The device you might be looking for is an analog mux or switch.
You can use the gpiozero library in python to interface with it.
You can do this with:
from gpiozero import MCP3008
pin_which_i_am_looking_at = MCP3008(<your pin number here>)
value = pin_which_i_am_looking_at.value
As far as I know, this should work.
Have fun with your MCP3008!
Ideally you want to load in 16 bits of data using the SCK, then output all 16 bits in parallel using the RCK. Binding SCK and RCK together makes normal shift / load operation neigh impossible. Normally OE/G is low so outputs are always ON. No need to turn off unless another 74LS585 pair would be driving the same 16 outputs
If all 16 outputs need to maintain ...