I'm sorry but there will be no reduction in the overall income gap with a raise in minimum wage. You admit the minimum wage increase will cause the jobs near that rate to increase as well as the medium income levels. The high end of the income scale is so far "stretched" as you say and held by so few people that the effect of the minimum wage increase is insignificant. Infinity -1 is still infinity.
Of course there will be the resultant unintended consequences such as the price of housing rising due to more people having a little bit more to spend, seeing as housing and healthcare are both increasing much faster than incomes and inflation it is likely these two will see the windfall and end up resulting in the lower classes having even less disposable income.
Of course there will be the resultant unintended consequences such as the price of housing rising due to more people having a little bit more to spend, seeing as housing and healthcare are both increasing much faster than incomes and inflation it is likely these two will see the windfall and end up resulting in the lower classes having even less disposable income.