Rudy, I don't have a weak arm, just a support arm..lol :)

I have right handed baitcasters also, got them at a garage sale from some guy, (RICK). I use them and do the switch over and have tried the left hand cast but i'm much more precise with my right hand cast. I don't need to cast like my daughter and throw the darn thing in the water and make daddy go get it.

All of my spinning reels are still left hand reel.
I had also contemplated the fact of one side setting the hook harder than the other, could work both ways though, I might set it too hard or not enough if on my support side.

I also broke my right arm when I was a child, I was not fishing back then but I did learn to write and do everything with my left arm.