I am trying to build some logic into an old-style game controller.
The controller normally works simply by a series of manual switches letting its respective pin be grounded or in open circuit.

What I aim to do is replace the manual switches with output from an arduino. However, I realize I cannot just connect each signal pin to an arduino digital pin, so my first thought was using transistors or mosfets. I read that transistors might introduce a significant voltage drop, so perhaps small mosfets, such as BS170, would make sense? Am I thinking along the right lines?
If I had to use relays I wouldn't really be in doubt on how to hook it up since they basically just open or close the circuit like switches, but it isn't obvious to me how I would connect mosfets to act in the same way. Looking at examples of how to use mosfets, I suppose I would connect the gate to an arduino output pin (with a 10K resister to ground in between?), but the other pins? The mosfet examples I find are mainly concerned about driving a load (LED, etc), whereas what I want to do is connect something to ground.
Maybe I am needlessly confusing myself? Or maybe mosfets are not obvious way to go?