Now that I knew I needed to make sure I had spaces for the vias under U1 (the microcontroller), I (eventually) deleted most of the traces and started rerouting a new solution. I've also used 0.2mm traces as default instead of 0.16 and it was still mostly routable. I've also tried to use 0.6mm vias where possible.

I've removed the USB related resistors and added a recommended 0.1u capacitor for U1.

I have also found a nice looking pad design for C0402s, used these recommended values for 0402 components and then made a similar one for 0603s. The silkscreen is much cleaner now. I should also note that I've made sure that the IC text is facing up for all components (which is the reason U2 is rotated 90 degrees).

Next time, I need to start thinking about creating some FPC pads so that I know the limits before I finish routing the remaining MP6543 pins.