As an outsider, here's what I am seeing.
In a way you did us a favor with signing JT. Even though it would have been okay to see how Trotz would have handled him, there's been a flaw to him that is really getting magnified with this season. I'm a pretty good judge of people due to my career and I've always been struck by what I see as his low emotional IQ. In his days on the Island I never saw a person who was truly comfortable with the full range of leadership. He was far more comfortable with the physical side of that (work effort, offseason work, etc.) but terrible at the verbal side of it.
And I think that because of his early years (the junior hockey waiver, the acclaim and the accolades in his junior career) he learned to be too robotic in how he interacted with people and the media, seeking affection and approval instead of being willing to put himself on the line. He was never the type in NY to kick someone in the ass, which is necessary once in a while if you are an effective leader. The stats were there on the ice but there were some really bad habits he had here too on the ice. He never really learned how to play in the defensive zone at the level necessary for example. Compare that to Barzal, who is really an Alpha male verbally and in seeking out a leadership role, and who willingly took a step back last year to inculcate what Trotz taught, and look at him now. Even JT's attempt at humor with Barzal ('that kid who won the Calder') backfired as seen by how Barzal talks about JT now.
He came into a situation with players that we can honestly say are works in progress when it comes to maturity in Matthews, Marner, etc., and that's where his reluctance to kick it in verbally is hurting the team as captain. He didn't have to deal with those types of personalities on Long Island. Compare that to captains like Yzerman, Messier, Clarke, Potvin and today's contemporaries like Crosby, Chara, Weber and some others. One thing that should have been a warning sign is his Team Canada play, where he clearly relished being the supporting player instead of demanding a leadership role. There's something missing here.
For all the issues in the construction of the team (I think the toughness issue is very true, but maybe even more mentally than physically, although physical toughness does mean a lot to mental confidence in the NHL) you still need your captain to hold others accountable, even if it means losing some popularity with them. I'm sure there were a few of Messier's teammates over the years who secretly wanted to ***** about him, but wouldn't dare. I don't see JT as having that sort of willingness to lead.
If it were me I would see about trading Matthews (the Kadri trade was a disaster to me as he could easily handle the 2C in Matthew's absence). A few other pieces could go too and I'm guessing if you open space up you could make a run at Greiss as a FA. This is really a tough situation for you unless the Cap does go up a decent amount with Seattle and the new TV deal. If there were ever a time to take a step back to take two steps forward this is it. One little thing I noticed in so many games, from all your defensemen: how many times way they allow passes to get through them on 2 on 1's. Since Pee Wee, defenders are taught to take away the pass but I'd bet big money your team leads the league in letting up 2 on 1 goals because of that flaw.