I've found the back and forth over Paul Ryan, Barack Obama, and the GM plant in Janesville to be remarkably uninformative. But I think the generous construction of what Ryan was saying is that he wasn't making some specific argument about Obama, the auto bailout, and the particular plant. Rather, he factory that happens to be near Ryan's house is synedoche for industrial idling as a general phenomenon. So above we have rather than anecdotes, overall data on aggregate industrial production.
What you see here is what you generally see when you look at Obama-era economic performance. You see that business cycle events don't line up perfectly with the start of presidential administrations. You see that if you judge today to the pre-Obama, pre-recession peak he looks bad. But if you judge today compared to the post-Obama, post-crisis low he looks good. You see that while it's unfortunate for Janesville that their particular factory hasn't come back, most of the lost ground has been recovered.
What I think is actually implausible here isn't anything Paul Ryan said about Janesville per se, it's his larger implication that following Paul Ryan's policy prescriptions would have boosted industrial production. Ryan is one of the few members of the House of Representatives who's actually articulated an idea that would have had a clear impact on this kind of situation, and it was a terrible idea—calling for an interest rate hike in October 2010. Had we followed his advice, domestic demand for automobiles and other consumer durables would have tumbled while the pricier dollar would have made American exports and import-competing manufactured goods less attractive. That would have been a double blow to industrial production, essentially ensuring more closed factories rather than fewer.