Absolutely back him. Get someone else in now, and they will want to analyse everyone in tbe squad before making any changes. In other words, all the deadwood will end up staying at least January.
Also, a couple of things that people don't realise at all is that being successful as a manager is probably even more about opportunity than it is about ability. There are many fantastic managers out there managing clubs with a very short glass ceiling hamstrung by tiny fanbases and no money. People act like Alex Ferguson and Brian Clough could have made East Stirling and Hartlepool respectively European Champions. It's not possible - and trust me, in SAF's case, if it were plausible, he'd have stayed at East Stirling for life.
The majority of managers learn the vast majority of the management ability on their UEFA courses and the first few years of their career. Klopp isn't much better now than he was at Mainz. More successful, obviously. But that's about opportunity, not ability. And good managers often don't get the opportunities they deserve. Chris Wilder has done a phenomenal job, but will probably go down next season with Sheffield United. Yet people will very, very incorrectly assume that, say, Arsenal's manager is better because he finished higher in the table and has won an EL previously.
Please. If Poch wins the CL vs Liverpool, are people going to act like that he learned all the managerial acumen in the few weeks preceding the event. Yeah, that is when he learned to be a manager *rolls eyes*, never mind all the years beforehand (even then, 98% of what he knows he learned within the first two years of management in Argentina).
In short, no-one knows what you've got in Solskjaer. People like to pretend they know more than they do, but whatever. He struggled with Cardiff, but a proven experienced manager just just gone down with them. Give Pep or Klopp that job, they might just be able to keep them up, but I guarantee they gravely struggle - they've both had experiences that prove there's only so much you can do with and against clubs sith m massive financial/other disparities to yours. He needs help though, mostly because buying/identifing players at the very top end of the ability spectrum that fit in with the club and players already there (an underappreciated requirement - within scouting it's a big deal) is something he's never experienced and is tougher than it seems. A DoF would be incredibly helpful for him.