Sara, I don't know where you live, but in many places the screen built into the chimney cap is required by code. It acts as a spark arrestor (in addition to discouraging birds) and prevents roof and wild land fires. If your chimney is reasonably accessible, the build-up can be removed with a couple sharp raps of a wire brush handle or extension pole. Burning hotter fires also prevents creasote from blocking the screen. A significant buildup on the spark arrestor may be a good indicator that the chimney needs cleaned anyway.
With modern high effciency wood stoves, secondary air is used to reburn the particulate and volatiles in smoke. I have noticed that the accumulation on the spark arrestor is almost negligible since I started using an updated stove. Look around (web or stove shops) for a stainless steel chimney cap. I find your chimney sweep's advice to be unusual for his trade and I disagree. The cap will make your chimney last longer, preventing moisture, birds and downdrafts from commng down, and will let you operate more safely, preventing sparks and flames from being conducted outside.