The challenge is preserving ability for content to control all pixels; without it, the content ecosystem ends up developing single-purpose, generally crappy apps, which isn't necessarily a better thing either...
I'm not sure it is the only solution either - what about "secure attention key" type ways to get the system's attention (in this case the browser's), bypassing any content interception? For example, what if there was a key combo guaranteed to always bring in the browser UI, and typing that key combo was necessary before inputting any password field?
Alternatively, the reliance on browser password management could provide some security if it can be trusted to always work...