Getting back to the original OP's question. My answer is yes, it makes sense to carry a knife or some other weapon on the weak side or shall I say opposite side of the gun hand.
This was driven home to me in a training video I seen where the student was being confronted by a man wanting money from him, at an ATM. The ATM was located in such a way that the student didn't have an escape route, he was cornered due to the layout of the ATM.
The student, did not attempt to draw his weapon (strong side from under a jacket) until the robber was close enough to quickly step forward and grab the student's hand and arm before the student could clear leather. The robber (instructor in this case) grabbed the students hand with both if his to control and disarm the student. The student's other hand was free. This is exactly where an off side weapon could save the day, and one reason I often carry a 6.5 inch Bowie knife in an inside the pants holster on my off side. All it takes is very gross motor skills to grab the handle of that Bowie Knife, and deliver a devastating cut or several in a split second, that can sever tendons, etc, while the bad guy is still trying to control my gun hand.
This is the question the OP was asking. He wasn't asking if he was better off bringing a knife to gun fight, as per the 21 foot rule, etc.
In this training video, I think the instructor was trying to show the problem with letting someone get too close before acting, and obviously the defender waited too long to draw, but we can get into these situations so easy, because, among other things, sometimes it hard to evaluate the danger, and we can't just go pulling on everyone that says hello and gets a little too close.
