Transistors can be used to amplify signals, they are not limited to acting as binary switches. If you use analog rather than digital, using transistors in this way means you can replace however many transistors it would have taken for multiplying two n-bit numbers with just one; I understand capacitors can be used for accumulation, but don't know how many additional components that needs as I'm an electronics noob.

The reason we don't do this in general (any more) is that for long chains between input and output it has been much too difficult to avoid accumulation of errors. LLMs happen to be extremely resilient to errors like this, which is also why we can use e.g. 4-bit weights.