Steve, one advantage of the midnight start on summit day is that you miss a lot of the visible exposure on the way up because all you see in front of you is the next six feet of track in the snow. Never a chance to get scared until you're nearly on top.
Now, I'll admit that acrophobia is not one of my problems but I do have an appropriate respect of exposure. For the most part, you're on moderate to steep slopes, not precipitous dropoffs. Plus, once you get to the top, the crater is this 1/4-mile wide bowl with 200' rims, so it feels like you're well protected up there as well. The tippy-top of the rim is even pretty wide and gentle although you wouldn't want to go sliding off the outside. (The inside into the crater is a gentle slope.)
"Hanging ten" off the summit boulders on top of Whitney is a whole lot more airy (and scary) so I wouldn't put Rainier high on my list of scary-to-climb peaks. Yes, you do have to step over a few deep crevasses but they were mostly a foot or so wide and as long as you watch your footing on the far side, no problems.
As for dying on the mountain, well, the overall death rate has been pretty constant at one per customer. I'll keep doing the things I love and not worry about it. People die on the highway every day but that doesn't keep me from getting into the car and driving to work, either. You do what you can to minimize the risks (wear a seat belt, travel in rope teams on glacial terrain, whatever...) and get on with life!