Good discussion re: Hagelin vs. Connolly.
If we look at a team making a playoff run right now, I guess I'd still have to go with Connolly, because I think his ES (per/60) scoring is flying under the radar despite being phenomenal (especially if you factor in not having elite linemates). I think I'd be somewhat surprised if he doesn't score 25 next year in a top-6 role on some team next year, if he leaves.
But Hagelin (both against us and for us) has looked like an uber-"role" player. His elite speed & motor & high hockey sense can have the kind of impact that's maybe comparable in some sense to Wilson's high speed & elite physicality (before Wilson developed a scoring touch, which obviously moved him up a tier). At his best, he's like a Beagle with twin turbos and two extra gears.
So in practice, it probably comes down to which component a given team needs more. The current Caps team is pretty well balanced, but if you think of the earlier skilled but flawed Caps teams, Hagelin likely would've made a much bigger difference compared to Connolly, who would've been another Fehr/Flash/etc.
If we look at the overall picture, I think it's much more clear-cut -- Hagelin's age should be the deciding factor against him, if the Caps have to choose between him and Connolly. However, in the short term, losing Hagelin could probably hurt nearly as much. Caps' vets are not getting younger, and the league keeps getting faster... It's true that formally his role is easily replaced by somebody who can do 75% of what he does (e.g. Stephenson) -- unfortunately it's the remaining 25% that seem to make a huge difference.
Re: Burakovsky, it's hard to know what to think any more. If he has a weak/mediocre playoffs, he's got to be traded, right? But if he performs "well"... I don't know, it depends; I think in this case I'd likely still choose him over Hagelin because of age, but probably he'd really have to be on fire to convince me that he'd be worth letting Connolly go.
Regardless, Bura's increasing his value, so good for him and for the Caps.