Again about four-terminal MOSFETs and relays
I think what you are suggesting is that you use a MOSFET as the contacts of the relay, allowing current to flow in either direction. That is commonly done in VLSI design using MOSFETs. The key terms include &quot;pass transistor&quot; and &quot;pass gate&quot; or &quot;transmission gate&quot;. You might also find some optoisolated MOSFETs that will do what you want.
The problem with doing what you want to do is controlling the gate voltage so that the relay is &quot;open&quot; or &quot;closed&quot; regardless of what voltages are applied at the source and drain. That requires circuitry, and circuits consume power. If you use one of the optoisolators then you need to provide power for the LED. It is also true that every MOSFET will consume some power itself when passing current, because the resistance from source to drain is not zero.