So, on this front:
In Philly, the broadcast is the mouthpiece of the organization. They are one in the same, and though team ownership relationships are becoming more common (see: Fenway Sports Group), the Flyers wrote the book on this. Ed Snider was the first to start a regional sports broadcast network, the predecessor to what are now the NBC Sports local channels. The Caps and Sharks play their games on NBC Sports channels, but the teams weren't founded or owned by the same entity.
When you listen to Jim Jackson and whatever ex-player stooge call a game on TV, you are listening to management's bullet points (last couple years: TORTS + TUFF TO PLAY AGAINST). You'll never hear anything remotely critical of management or ownership decisions. It's more corporate PR than standard homerism. Jones moving from the NBC Sports Philadelphia booth to the team president role displays how things work at the cable conglomerate's hockey team.
The Flyers broadcast spent very, very little time on this. The pregame show started with the host reading the statement and immediately transitioned to talking about how Ersson was going to be tested. The much-hyped (supposed) future of Flyers goaltending just took a leave of absence during a playoff push and they might have spent 35 seconds on the subject.
They of course also mentioned it at the top of the game broadcast, and once again read the statement and instantly transitioned to talking about Ersson having a larger workload going forward. "A real test" and "a big opportunity" and "the team has confidence in Sam Ersson" were refrains throughout the night. A shot of Cal Peterson on the bench. They had a good game to broadcast, so it was easier to avoid. Once again, no well wishes or passing words of support to indicate a mental health leave.
In postgame, didn't get a ton of questions or answers from players. Hart's roommate, Cam York, was asked about it and mentioned that he'd miss him and that he's one of his best friends and that the situation sucks. Morgan Frost was asked about it and flatly said "I'm not gonna comment on that." Most other players weren't asked. A beat tried to ease Ersson into talking about it by asking if they fell down 2-0 in the first because "their focus maybe wasn't there" and he just ignored that concept and talked about how they needed to stay out of the box.
John Tortorella held a quiet 49-second press conference in which he answered one question about why they lost and fielded a softball "with Carter out, how confident are you in Sam" question by saying "Sam's played very well, totally confident in Sam" before walking away.
Carter Hart has been a mercurial keeper and has spoken openly about his extensive work with a sports psychologist over the years. A mental health situation, or any other type of personal hardship, is highly plausible. I'm just telling you how this is being presented by the team, which tonight was a very firm and consistent no comment. Compared to the hardcore press blitz the Flyers did two weeks ago in the wake of the Cutter Gauthier news, this was a stark difference.