A potentially great improvement to minimum wage laws to handle both economic efficiency as well as poverty concerns
Let's look at the actual intent of your article - it is not to propose 'the' solution to improve the productivity of the economy, but to device methods to improve it. After all, the fact that you say that you have not spent time thinking about the implementation (and from the depth of the article, the issue itself) reiterates my opening line. Firstly we have to remind ourselves that productivity of an economy is not even a non-linear problem, but a transcendental problem. So linear proposals like the one you attempt will barely scratch the surface.
For a moment, think of Minimum wage limit as analogous to minimum score required to pass a subject. In a class, if a kid has not learnt enough, he is graded 'F' . He may have scored 1 point less than the minimum required, but the point of grading him 'F' is, he needs to relearn (improve skill). Society stigmatizes him as an F-grade student, but the system only summarized the result of his action for that academic year. The school's objective is to maximize the pass percentage. Incentives should be methods devised to encourage students to improve learning (and only indirectly to improve scores).
Now, back to your article. Welfare and method you describe only have incremental effect on the unemployed because they tinker the net income earned by the 'unemployed'. At best it will move the equilibrium from one point to another, and only gradually over time. I would be careful to describe them as pareto improvement, optimal etc in such situations.
Perhaps if the mechanism you describe is deployed to incentivize skill improvement & corresponding mkt demand creation by the government (,rather than wage increase per se), it would have positive systemic effect on the state of economy.