>"One thing I don't hear a lot about is the fact that as nice as razor blades are, they cannot be resharpened."
oh, but they can be resharpened.
Try this on a shaving razor in a handle. It could be a $0.19 Bic or a Sensor 3-bladed model.
When it's gotten noticeably dull through use, hone the blade (stropping, really) on leather. Not full-grain leather, but on the back side or on suede. The back side of a leather belt works really well. Push the razor backwards - at exactly shaving angle, but in the reverse direction (you're honing the blade, not shaving the leather). Use moderate pressure. Repeat about 20 times. It takes about 10 seconds with practice.
Back in my Boy Scout days, I'd always finish any knife sharpening job by stropping the edge until I could shave with it (by wetting arm hairs with spit and checking). Nowadays, I'll do it in a pinch, like when I'm on a business trip and my shaving razor is a bit dull, I can get back close to new.
Edited because Nick and I cross-posted: Yeah, and if the razor blade had a perceptible nick in it, you could use a fine whetstone or the UL version: a patch of fine emery cloth on a flat surface. For coarser work: 2-3 inches of 1-inch wide plumbers sandpaper on a flat surface. Cloth-backed, water-proof, designed to sand metal.
Yes, HYOH, CYOBlade (or not). I'm cool with you carrying a fixed blade, I doubt I ever will. I have some admiration for those who go all the way to a single-edge razor blade. I'm happy in the middle with my SAK Classic.