to button the cuffs first, it’s so much easier for me to fasten the right-hand cuff buttons using my left hand, then the other way around? Do most people in this situation find it easier to button the preferred-side button
with the non-preferred hand, or is this an idiosyncracy on my part?
If your ploy was to get me to unbutton, then rebutton my cuffs, then you succeeded. 
I seemed to have had equal success at both the left and the right cuff.
I wear mostly short-sleeve shirts, but I know from experience that my wristwatch causes me problems when I try to button my left-side sleeve with my right hand. In addition to adding more circumference and thus making the fit tighter, it also gives my fingertips something to catch on while manipulating that tiny little button.
Hope this helps.
Right handed, do it every day, tested it several times:
A draw.
Ditto what Philster said. I think you’re unique, javaman.
I don’t notice much difference between hands when doing up buttons (although I usually have to adjust my wristwatch on my left wrist). However, I do find that it is easier to affix my right cufflink when I’m wearing French cuffs than my left. Never really thought about why, but it is something I’ve noticed in the past.
I have a hypothesis (though, as I said, I don’t suffer this problem):
Hold your hands up. This is the position mine are normally in when the cuff is being buttoned.
Look at the orientation of the buttons/buttonholes for each cuff, and compare to the central buttons on your shirt front.
I notice that my left cuff is exactly like my shirt: the button is sewn onto the right (underneath) flap, while the buttonhole is on the left (top) flap. So, in theory, buttoning the left cuff is the same as buttoning your shirt, which should make it easier than the right.
But you’re backwards, javaman.
Do you, by some chance, hold your arms down/straight out rather than holding them up when buttoning? This would reverse the orientation of the buttons/buttonholes, thereby making the right cuff more like your shirt, the left cuff opposite.
I can’t believe I went through all that.
Actually now that I think about it the difference in difficulty only if I hold my left lower arm roughly at stomach level and parallel with the ground. If I bend my elbow and hold my left cuff up, it’s much easier. But the
right cuff is still easier even then.
I’ve heard that some people are generally right or left handed but use the opposite hand for certain things. Paul McCartney is right-handed for everything else but playing the guitar or bass, which he does left handed. My case must be something similar.