Buck Converter won't switch properly
I used TI's WebBench tool to calculate component values for their
  reference design, and had a board fabricated according to the design:


Allow me to express a serious doubt about the board fabrication statement. Here is your layout, and the recommended layout, side-by-side:



Their layout is done that way for a reason, and the reasons are explained in many places, e.g. here. All MHz-range switchers are fundamentally sensitive to implementation of "high-current" loops. You can't get decent (advertised) efficiency (nor even basic functionality, as this example shows) without reasonable implementation of these loops. Especially in a 10:1 design. Manufacturers spent a lot of application resources to make right layouts, and design of internal pads on IC are usually optimized for particular fan-out. There is no reason whatsoever to deviate from the reference designs unless you can propose a better layout (sometimes it may happen). In this design the most glaring mistake is the routing of high-current switch node (around SW pin). Huge parasitic inductances of skinny traces likely change the switching curve of internal transistor, which likely causes some odd internal behavior and refusal to start the switcher properly. 

Also note the tight low-inductance placement of Cin and Cout caps. And the designs usually use no thermal relief, only solid connections, all for the same reason - to exclude any parasitic inductance in high-current switching loops.

I am afraid this construction can't be salvaged and needs total re-design, preferably in full accord with manufacturer's suggested layout.