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The color scale hugely amplifies minor differences, see my other comment. In the dataset, 15 years (2000-2014) are represented and the weekly cycle is therefore present. This could have been mitigated by using a 28-year dataset. Here is how often each month started on a given day in the dataset. We don't have colored text so I used emoji.
Here is how likely any date is to be Monday-Friday in the data (out of 15):
You can see for example that August 01/08/15/22/29 is 10-20% more likely to be a workday than other days, which corresponds to the extra births. Feb 29 cannot be read directly from this table; its value is 3 out of 4 (2000: Tue, 2004: Sun, 2008: Fri, 2012: Wed).
As for your question: every month has approximately the same strength of the weekly cycle, it's just that the prevailing colors in July-September show them the most. August's is more visible because it does not have the disrupting July 4 and September 11.