Could this be improved by a government agency where they do this paperwork for you at a per-employee cost if your company is under X employees (say 3x the limit at which the paperwork kicks in)?
The easiest improvement would be to remove the employer from the equation and just provide taxpayer-funded benefits to all. The main reasons employers are still involved are inertia followed by a concern that employees won't feel as obligated to stay in shit jobs.
Sounds like socialism, though. Can’t have that!
I think there are arguments to be made in favor of a more free market approach (or a less free market one.) At this moment we seem to have the worst of both worlds.