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The high period before/after Christmas so that more of the medical community can have Christmas off is pretty strong.
I have no idea why August has such a strong weekly cycle, when July/September aren't that clear. I mean... surely weekend vs. weekday is canceled out by having multiple years that people are born in. If someone knows, I'd love to hear the answer.
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.