So I did the replacement according to this guide, with a replacement fan purchased from iFixit, and the fan would not spin even under heavy load! I took out the fan, and noticed that there was quite a lot of resistance when I spun the blades manually. It turns out, the fan was slightly squashed flat near the vent so that the resulting friction prevented the blades from spinning free. I un-squashed the fan, re-installed it, and the fan is spinning! The moral of the story is to make sure the blades can spin free when installing a new fan.