I think she used a poor choice of words to translate President of the Senate into plain speak...as "in charge of".
Constitutionally that's the ONLY defined role. Meaning her critics saying she doesn't understand the constitutional role are the ones that are incorrect. If they came back with she doesn't understand the current role, that's a better criticism...but you can't determine from the one sentence of that indeed was an issue.
IN ADDITION, since you bring up coaching...aren't all her critics saying that she refused all coaching? (part of the whole not knowing NAFTA and Africa is a country furor)? So if she wasn't paying attention to the coaches, maybe her debate answer is actually her own?