I agree that a good doppelganger wouldn't even bring up the possibility of a doppelganger, even if he's pointing them somewhere else to look for one. Why even set the notion in their head? It'd be safer to try and throw them off track some other way. That's just one more little fence in the way of them discovering the truth. Plus, it's probably a violation of some sort of unspoken doppelganger code somewhere.
Besides...the very first thing I think I'd ask when provided an answer like that is simply "How do we know that you're not the doppelganger and the one we met was the real one?" And that's obviously not something the doppelganger wants to be asked to prove.