@WreckingCrew is probably your best bet to answer. Most of us only watched a handful of regular season games, then the playoffs. But I want to say the organization was very much on the same page coaching wise, so Brindy and him are/were very similar: skating, skating and more skating with hard forechecking/backchecking based around puck possession/puck support. Vellucci might change it up a bit more to his liking in a new organization though.
That said, that seemed to be what Sullivan was all about during the back to back Cups, so I'm not sure how much Vellucci would change given the same players. Granted, some coaches have a short shelf life so who knows.
I think that's a pretty solid description, the team really thrived most at counter-attacking I think. They'd prevent as many high-danger scoring chances as possible, force shots to the outside then use their speed to break out up ice to try and get an odd-man rush. It definitely helped that we had a pretty vet defense and fast forwards like Necas, and Geekie in the playoffs was just relentless creating chances. Once in the zone, keep it there at all costs...plenty of nights we'd just cycle the puck for extended periods of time hoping they'd tire or get out of position while changing up 1-2 of our own at a time.
There were plenty of nights where the team overall kinda played like garbage (had trouble connecting on simple passes), BUT when they did connect it would often lead to a scoring chance...so they could seemingly get outplayed and still win. Also, one thing under Velluci I haven't really seen before (Canes this year were similar), was the ability to come back in games...even down a couple goals the team was rarely completely out of the game. Sometimes it was too little, too late, but they'd sure be swarming. He was pretty good at keeping them in most games and helping them shake goals off.
I think our PP was pretty average (but in all fairness we were one of the worst at home for a long time), but the PK was stifling! We had like 5 SHG against TOR and CHI (maybe 6) in the playoffs, and TOR was 0 for 8 I think in game 6. No matter how talented the players are, the system he had for our PK was excellent (86.6% regular season, 2nd was 85.1% for reference).
We've definitely a number of guys unexpectedly develop that many had doubts about (Gauthier & Bean have noticeably grown while he was HC). I have no doubt he's a part of the reason so many guys on the Checkers look like they could have a future in the NHL.