I've got two Polytruder extruders and they both have a bunch of Dayton blowers on them (small 110V 7A FLA units) for cooling the barrel. I've found that these blowers are each running through GMA 10A fuses. These are wee little 5mmx20mm fuses, and more importantly they are fast-acting fuses. This seems odd to me. I've decided to replace them with GMC 10A fuses for a wee bit of extra inrush protection, but can anyone think of any reason why these GMA 10A fuses would have been used in the first place?

They shouldn't have been fast-blow, and that may have been a factory or maintenance mistake since those tiny fuses look pretty much alike.
Your fans are probably 1/4 to 1/3hp split-phase or even shaded-pole motors, and they trade simplicity and cheap cost for a fairly hefty starting current and some inefficiency. We always run these on slo-blo fuses or thermal-magnetic breakers. If run on magnetic-only breakers, the instant-trip point is usually set at 8x FLA or so. Your fuses are only 1.4x FLA, so I am not surprised they won't start the motor.
Try the slo-blo fuse and tell us what happens.

PS - we have a similar extruder here, and that barrel can get pretty hot even with the fans on. How close is the fuse block to the barrel heaters? Your fuses may be starting out with a thermal disadvantage...