Both inputs connected to both inputs of a NAND gate? This sounds like a short circuit between both inputs.
After all you don't need any gate for that kind of "OR gate".
In practice connecting the inputs of a NAND results in a NOT. And having two NOTs in sequence is ID. It does nothing (only in a logical way, you can use this construct for different things in reality).
The problem is, that the OP's train of thought is only working with the switches drawn in the picture. In digital electronics you have a signal as either high (e.g. +5V) or low (GND, 0V). And you're not allowed to connect two such signals without anything between them.