D can't do anything if their own F don't back check, and if that's the case, they are essentially outnumbered, and the goalies gets hung out to dry. That addresses transition/rush.
Once the other team has them hemmed in, the oilers are stationary - good that they have orange in their jersey colors, since they are pylons. Allowing obscene puck movement/cross seam plays that again leave the goalies hung out to dry. The remedy for this is tougher, relies on D/C reading each other, Wingers knowing when to (briefly) collapse before hustling to reestablish coverage/pressure on the points, but ultimately requires them to always be in motion, even if it is on the same tiny patch of ice. Yes, that can be done, esp using rotational energy and translating that into straight line movement.
Need a coach that understands and can communicate this, and complete buy in by all players on the ice at all times.
Changing goalies or D is a patch that will have very limited success.
Coaching/systems/Buy-in, and then whatever whenever to upgrade personnel.
Their GM sucks, and unless McDavid takes a big discount, he will never win a cup as an oiler.