The ganseki otoshi you mention looks like a more deadly version of kata guruma, shoulder wheel.
I have been able to do the perpendicular hip koshi for about a year now - and a good thing it is too - I haven't been able to bear weight on my right leg for many years. In that I'm healing now, and I can do the parallel hips type koshi nage again too!
I think they are both tools in the aiki toolbox, and I think perhaps we see the great masters doing perpendicular method because they are older, and the 'harder timing' perpendicular method is a damn sight safer and less effort for them.
PS - Throwing uke up over your head and finishing holding their arm as though it were as a sword is really really fun. As a teenager we trained in a second floor ymca - we threw such that uke landed between, not across the joists - a really good throw saw uke bouncing, sometimes even back to his feet with a little help from tori. gooood times.