How well this could work depends highly on what type of business you are in. In my industry, a company is expected to be available AT LEAST during normal business hours.
If you were in a business that was structured differently and had less required interaction with other businesses then you might have a better shot.
Another down side is that a company has a certain amount of fixed costs and overhead (accounting, payroll, taxes, benefits, etc.). So, it's less efficient to pay more people to do the same work.
On the bright side, you would have very little competition for the top employees who wanted something like this so you could potentially have very good retention and morale.
At my office, this is how all of the top people work, and we're all self employed independent contractors, so there is no payroll, benefits, or taxes to worry about. We all handle that for ourselves.
We share the workload, cover each other's vacations, and balance out the responsibilities among multiple part time professionals.
There are many many organizations out there utilizing a contractor situation where staff work flexible hours and organizational overhead gets minimized.
The problem is that there are so many protections and benefits for employees and virtually none for contractors.
As the gig economy grows, the system will have to adapt in order to accommodate this kind of organizational evolution.
It's not likely going to be that the current employer/employee structure adapts to part time professionals, it's that the part time independent professionals will become so ubiquitous as to require their own system of benefits and protections, and the employer/employee structure may then change.
