Those raspberries I left were a bit of a tangle once the snow left this spring; I'd run out of recycling bags in the late fall, and cut down only half of them. The ones I didn't dig up were also the same ones I didn't cut back last year. Lo and behold, they are now COVERED with flowers and forming berries!
I think I've figured out what happened. The late-bearing raspberries are cut down to the ground every year, and they come back each summer with new canes that bear fruit. But these other raspberries must bear fruit on OLD wood. I was cutting them to the ground every fall, just like the others, and so was eliminating all the fruiting canes; just a few would struggle up, probably from stumps left behind, to produce a few pathetic berries. I think I've discovered the secret now; once they finish fruiting, I'll cut the older canes, and leave the new ones to produce next year.