It is just as much the scheme as it is the players.
A hybrid 3-4 is not a 3-4 defense but they should incorporate a few fundamentals. It could benefit them. However, in order to do that, they would need legitimate LBs and not converted EDGEs.
EZE is not a LB. Rashan Gary did not truly emerge in Green Bay until they switched back to the 4-3 and made him a DE.
These former EDGEs and other like them lack the versatility to perform as complete LBs. There are times when a LB should be called upon for pass coverage. Both their own coaches and the opponent coaches realize this isn't going to happen.
Therefore, they will be used to pass rush in most cases and the opponent will know this.
This effectively eliminates one of the most important advantages of a 3-4 defense which is the direction of the 4th pass rusher. In a traditional 3-4, the identity of the 4th pass-rusher is unknown. He could come from anywhere, a LB, a safety, even a CB. This confuses the opponent O-line by creating a degree of doubt.
It is far easier when they know who the pass-rushers are and this eliminates a key advantage of a 3-4. If Rashan Gary is on the field during an obvious pass play, the Cowboys could put him at safety and it wouldn't change the opponent's blocking assignments.
The Cowboys must diversify their 4-man pass rush in the same way as a traditional 3-4. Their 3-man front is talented but must stay conservative in their lanes. Again, knowing the identity of the 4th pass rusher give the O-line the advantage.
I think EZE is now in the unfortunate position of playing on a defense that is not compatible with his skill set. Rashan Gary is also better suited for a 4-3 defense but they could try him at DE where he would still be considered smallish.
Trading for an elite 4-3 pass-rusher is not going to solve the issue and would probably just create more of them.
I have to question some principles of Parker's defense when it claims to be an innovation of the 3-4. They need the right kind of LB with the right kind of skillset. Parcells knew this and always stated that you could never have too many good LBs. How many do the Cowboys actually have?
In the end, it will take some personnel changes for this defense to become elite. Considering the draft capital available over the next few seasons, this will be another challenge.