Our team doesn't really have a department per say - it's clear to me that we are behind most other teams - however, Brad Holland definitely uses them and wants to invoke them more, if you hear his interviews.
Would especially make sense in the Kulak, Ekholm additions for playing a role.
It also makes sense that they likely paid attention to analytics when trying to still re-sign Puljujarvi before the deadline - it's likely he didn't accept more because Jay Woodcroft had decided not to play him and he didn't want the 13th forward role on the team.
I'm not sure why you are so sure about the state of the Oilers analytics work or that within teams across the league. Or how to gage the role and actual impact being made within any organizations. Cherry picking Kulak and Ekholm as advance stat based trade decisions is a stretch with no basis. Pretty obvious deals.
A list of analytics/data-driven roles across NHL teams:
NHL Analytics | Behind the Benches
Look at bit closer at one by way of example Dallas Stars. Their assistant GM is a communications guy by training and worked as a media relations staffer. His bio and scope of work is contracts, scheduling, CBA and cap, mental health, oh ... and analytics
The official National Hockey League website including news, rosters, stats, schedules, teams, and video.
www.nhl.com
"In his current role, Janko is primarily responsible for player and staff contract negotiations, and has played an integral role in the construction and framing of the contracts for several of the team's core players including Jamie Benn, Miro Heiskanen, Roope Hintz, Esa Lindell, Jake Oettinger and Tyler Seguin. Additionally, Janko oversees all preseason and regular-season scheduling, Collective Bargaining Agreement compliance, cap and budget management, salary arbitration, strategic planning and works closely with Dallas' scouting staff, with an emphasis on the professional level. Along with Jim Nill, he created and now oversees both an analytics department and mental health division and also serves as the team's liaison for all player transactions with NHL's Central Registry.
Janko, 46, joined the Stars in 1999 after serving in the media relations department for the Mighty Ducks of Anaheim during the 1998-99 season. A 1998 graduate of the University of California at Santa Barbara with a degree in communications, Janko is a native of Mission Viejo, Calif. He and his wife, Jessica, their daughters, Reese and Hayden, and son, Carson, reside in Highland Village, Texas."
Well down the org. chart are two scouts/analytic coordinators, one the son, of the team's CEO:
Scouting Staff | Dallas Stars Hockey Operations
NHL organizations work with external data companies which probably is the most efficient way to breakdown walls of information from a high speed, chaos based sport. I speculate the internal quality of brain power doing analytics based support work likely varies dramatically as does the actual impact delivered.