Most non-Golem constructs are harmlessly powered by pure magic.
Golem magic immunity is my least favorite rules argument. And that they're unnecessarily powered by elementals is next in line. I like to think that the folks who made all this up to begin with just didn't make the connections between sapience, sentience, and free will. That non-humans were made to be non-persons by the rules themselves never sat well with me. I suspect that's part of why i favor playing "monster" characters so much myself.
Even if we could be so specific then you're still saddled with a minion who has a mind and free-will and alignment entanglements. Generally speaking it can be difficult to do away with all three of those factors as thoroughly as you can when commanding a Construct.