This suggestion is good for two reasons. First, real-world tasks such as programming, starting a business, etc. do require an ability to perform drudgework, but don't intentionally introduce it. By creating something of value, one demonstrates the ability to buckle down and deal with setbacks far more convincingly than a monthlong program ever could. At the same time, one demonstrates ingenuity and drive. Second, the initial idea lacks one very important factor: "how do we get there". A good idea can't just be good once achieved. It must also contain a reasonable path towards its achievement. It seems rather unlikely that most businesses would want to hire the graduates of a monthlong addition program for reasons described by many other posters.

Innovation has one disadvantage: it is risky. A failed attempt will look a lot like slacking, and provide negative signals to potential employers.

A third suggestion that lies between this one and the OP's would be to find a job immediately after school. Traditionally, most such jobs are for less-intelligent applicants. It would be reasonable for a high-status employer to create a program that hires high-SAT highschool graduates to work on marginal projects at a low salary for a year, with the understanding that those who performed well would receive not only experience but also a good recommendation for future employers. The benefit to the employer would primarily not be direct profit, but rather a pipeline to hire the best of these intelligent students.