I don't think Edmonton does this even with their need for a goalie. They also need to improve their defensive play on the blueline, get tougher, and work harder, and this trade makes Kulak a 2nd pair D again, downgrades a lot on the wing for a player who likely will ask for a big raise in the summer.
Change Pickard to Campbell and it gets more palatable for Edmonton. They are still losing the 2 most valuable pieces in the deal (Swayman is great, but goalie value is weird) but at least they get rid of Campbell.