The kid went through a sniffly, congested phase a week ago, so be rolled out the needle-tipped squeezeball nosepicker to help her breathe. A couple of awkward attempts caused her to go into meltdown; and very soon, she'd start screaming like a banshee as soon as the bulb came into her line of sight. If you add up vestigial childhood/infant memories of a) suction nosepickers and b) rectal thermometers, I bet you could explain about 90% of all the alien abduction claims out there.
Anyway, here's what I noticed: I was prepping the nosepicker too late. I was squeezing it flat (in order to suck up the snot) right in front of her face, which sent a concentrated jet of air right up her nose. THIS is what freaked her out.
Once I figured out to get the thing ready on the side, I'd just talk to her calmly, explaining what I was going to do [a la Charlie Brown's teacher: "mwah mwah snot mwa mwa mwa"], she was totally cool with me poking a rubber syringe into her nostril.
Oh, and maybe you're much better at it than I am, but I never got much out. The squeegee seems effective at bringing the snot to the front, where you can get it with a kleenex or the elongated nail on your pinky. ["Why no, officer, it's to pick my son's nose."]