Man, I totally get where you're coming from. I mean, painting before you put it together seems smart for the edges, but @danielhenderson is right, you always miss spots behind supports or in corners. My go-to is to do a first coat on all the pieces, assemble, then use a small foam roller to hit all those impossible spots after. It saves you from that awkward brushwork flat on your back.
Painting before assembly does make edges easier. But what about the spots you can't reach once it's built, like behind brackets? Do you touch those up after?
That "spots you can't reach" thing is exactly what messed up my buddy's project. He painted everything after the install, totally confident. Then he saw all the bare wood behind the middle shelf supports. Had to try and brush paint flat on his back, got dust and drips everywhere. Looked pretty rough in the end. He swears he'll never do it that way again.