I originally learned irimi nage by not allowing uke to rotate into nage. This resulted in the "backward throw" that somewhat resembles the clothesline throw from WWF (or WWE, or whatever). It is not difficult to imagine stacking our partner's feet over their head and going back-of-the-neck into the ground while stacked. It was spectacular if uke was prepared to prevent her feet from stacking (because uke spun around nage's arm like a cartoon, feet flying and everything). But it is a very uncomfortable throw otherwise.

Here is a a sample from Shioda Sensei. Check out the 1:15-1:25ish mark for some irimi nages:
http://youtu.be/kj0TgZTs2cg
I understand that these were not comfortable ukemi. Shioda sensei's irimi nage is also a good look at what the throw looks like when you reduce the "up/down" that we see exaggerated in other demonstrations. They tend to look more like a Hulk hand slam...

Then came along the variation that allows uke to rotate into nage and around the hip (or pelvis), which I believe to be at least a little safer, if not more comfortable. The change in rotation point gives you some ability for a little more "umpf" without necessarily throwing uke on her neck.

Diagonal movement using the imagery of corners is pretty common instruction in training for kata. Understanding that at some point, yes, you should transcend the need for those "mechanics". The gi grab should not be the technique, but rather to control uke during the technique ( I believe due to the exaggerated movement) - it's more as a courtesy for uke...