As someone whos knows almost nothing about 3d graphics and its APIs, how does Vulkan play into all of this? Why should this new API be used instead of adding Vulkan's API to the browser?
In general, Vulkan was not designed with the web in mind. For example: Vulkan's design intentionally has a whole lot of undefined behavior. Instead of strictly specifying what happens in all possible accidental cases, they strictly specify what is defined and provide debug layers that help you identify during development when you are unknowingly stepping into bizarro land.
Also, I expect Vulkan was designed with the expectation of process-level isolation being the end-all of security concerns. That's not sufficient for single-process, multi-tab browsers.
Chrome has process isolation, and Firefox is developing process isolation right now. I don't think it makes sense for a new technology stack to go out of its way to support single-process multi-tab browsers.
It does allow you to create pipeline state objects ahead of time like Vulkan and D3D12 but they only include a subset of state (shader + antialiasing state + color/depth attachment state). The rest of the state is set when recording the command buffer.