As far as I know the only options are manually tracking (time consuming), data scraping (limited), and video tracking (don't know how advanced/accurate/useful it is now)
Not to keep beating a dead horse but idk why no one ever asks about this. Not expecting them to reveal all their secrets but none of the media even makes an attempt to get even basic info
Similarly to Tsujimoto I'd be surprised if all teams doesn't subscribe to some of the data-providers available.
SHL has a contract with SportLogiq which is an AI based tracking tool. On Swedish television they have shown some of the data that is available, for the players they get an email after each game showing all their shifts, for the analysts, it's essentially everything from xG models, to zone entry, zone exit data, passing data, you name it.
I would guess all NHL teams also have this information.
If you listen to PDOcast Dimiti occasionally has a goaltender-analytics guy (don't remember the name) who has access to this data as well and he talks very highly about the amount of information they have compared to what is publicly available when it comes to shot location, shot speed etc.
For the NHL teams I don't think it's a lack of data available, but rather how integrated the Analytics is in the day-to-day decision making.
As for Sabres, Lance Lysowski recently talked to Ranting Ron, where he mentioned he knew Housley used analytics because he talked about changing a tactic based on how a specific team plays (although I wonder if that's more analyzing video, rather than number, just relaying what Lance said). Whereas he had never heard Krueger mention analytics.