That being said, it looks like the capacitor is only there to prevent power surges from the power supply.
Exactly the opposite. It holds up the supply voltage in the face of rapidly changing current demand. This is caused by the PWM or switching on and off of the LEDs.
See the power and ground coming out of the end of the reel? Connect this to the power and ground you are already using at the other end of the reel. That will help with power distribution.