look who showed up at his event protesting against Kavanaugh:
https://www.theguardian.com/us-news/2018/oct/01/jeff-flake-brett-kavanaugh-latest-boston-protests
and, yes, he thinks he’s running. And he think’s he’s running the investigation - he has to ‘make sure’, the implication meaning McConnell etal need ‘his’ special supervision (same link):
“Flake told a crowd of young entrepreneurs attending a Forbes “Under 30” business summit that his staff is in contact with White House counsel to make sure the investigation into accusations of sexual assault against Kavanaugh is “up to standard”.