I have to brag, but first I have to tell on myself a little bit. You should really read the datasheet more thoroughly than I did. On the DRV8304 gate driver in 3PWM mode, the INLx pins need to be connected and pulled high for the phases to be turned on. If the pins are left unconnected or pulled low, the driver will put the phases into coast mode (all MOSFETs disabled). Also DVDD is an output pin, so don't connect it to 3V3.
In this image you can see where I cut the trace from 3v3 to DVDD (between the C and the 5 of the C5 reference). Happily, I was able to scrape some soldermask off the trace before the cut, and then bodged some 34AWG magnet wire onto it and connected it to the INL pins to pull them high. After this (and some fixes to a couple of failed joints on other pins) the device was showing correct outputs on the phases.
This is the first time I've ever bodged a PCB so I'm really excited I was able to get it working. This is just a test board for a more complete project I'm going to do down the line, so i'm not too worried about the longevity of this fix. But it's good to have this skill in the toolkit.