Yep. It should. From the guide:
Wait three seconds for the MiniPOV to 'boot' and you'll see the red, green and blue LEDs light up in order. Then the LEDs will flicker white.
The LEDs are basic RGB LEDs, with each color (red, green, and blue) controlled by a separate signal. By seeing only red, it indicates the possibility of a signal issue with the other two colors. There are several solder points that could cause this. If you post a second photo of the current solder job I can try and see if any of those specific points look like they might still have an issue.
The idea with plugging it in with a USB cable is just to provide another check.