Because it didn't work with Barrie in TOR doesn't mean that it can't work, and this is the perfect example of confirmation bias. You make it sound like getting Chychrun would be an easy task. The trade price for Chychrun is going to be VERY high. He's obviously would be a GREAT add, but because of that I'm not sure if the Oilers can (or at least are willing to) pony up the necessary assets to get that deal done.
In the short term - that would be the direction I would go. I don't think that adding a SAH home guy is going to fundamentally change the Oilers team structure or ability to defend as a team. How do you think that Luke Schenn would look on the Oilers? 15 of the 16 teams which make the play-off's end up disappointed. It would be a "big balls" type move, because it's hard to defend if the Oilers fail and you don't follow the conventional wisdom like "You win with defense". While that is true, I just don't think that the Oilers can change the way they play with 25 games to go, and they're better off trying to outscore their mistakes. YMMV.
EDIT: And for those saying the Oilers should concentrate on defense (which I agree with in principle) - that is pretty damn obvious. I would ask that if it's more likely that the coaching staff hasn't tried that, or if the current roster just hasn't been able to make that work? And if the current roster hasn't been able to make it work, does upgrading Kulak to Schenn (or the equivalent) really move the needle?
It’s not confirmation bias when the situations are very similar. Edmonton is front-stacked in forwards in mcdavid, draisaitl, rnh, kane- so is toronto with nylander, marner, matthews. Oilers have bouchard, barrie, nurse. Leafs have muzzin, rielly, and barrie. Both teams have questionable goaltending (larger sample size is needed for skinner though). Stanley cup winning teams have a robust defense corps. They need a minutes muncher. The closest thing they have is nurse, but another one would greatly improve their team. Expecting to increase offense with the addition of an offensive dman is dangerous because he can cannibalize production from the forwards and diminish returns.
Contrary to what you think, the oilers do have the assets to make a move. They still have 2023 first, 2024 first, bourgault, bouchard. Oilers can also target orlov who should be much cheaper. Someone will be available for those assets.
Like i said earlier, if you don’t have the right coach, just wait until he gets fired. The new coach can come in with a better d corps than with one that consists of all offensive dmen.