Please move past the language argument. Language is clunky and imprecise. You know the meaning and intent of the words I'm saying. If there is something god cannot do, then god does not have unlimited power. 

As you said, "god would be restricted to an extremely large but finite set of powers." Ok. I agree. Assuming there is a god, the range of powers is large but not infinite. There exists things outside of the power of god. Therefor god is not omnipotent. Extrapolating from that, if one of the main pillars of what god is, is called in to question, then that calls all the other descriptions of god into question. If we have now found limitations to god's power, where is that line? What powers does god have, and what is outside of god's abilities?