I don't know if anyone can give the "correct rationale" for the peaks, but we can make some educated guesses.
I agree with your explanations for the rise and peaks in the 1870s/80s and 1940s. The Contagious Diseases Acts were quite controversial since their passage, and the press for their repeal gained quite a lot of publicity. Additionally, penicillin was a major breakthrough in treating a number of different diseases. I'd also argue that Nelson and Mayer's creation of the Treponema pallidum immobilization test (TPI) in 1949 may have played a role in the 1940s peak (see Tampa et al. (2014)).
I also agree that the early developments regarding Treponema pallidum partially caused the early peak in the 1910s, first with its discovery in patients with syphilis and then the link between it and neurosyphilis, as put forth by Noguchi in 1913. There were other major breakthroughs around this time. You mentioned that Salvarsan and Neosalvarsan were major treatment options, thanks to Ehrlich. However, its application to war may have been what resulted in the peak. LW Harrison wrote quite a lot on this. Furthermore, developments in the use of bismuth (see here) in 1916 and then 1921 made it a second option, replacing arsenic. Without World War I, Ehrlich's discovery might not have gained quite as much attention. Also, the Wasserman test, developed in 1906 contributed to the diagnosis of syphilis.
There's one more amusing piece of information that may have led to a data bias in the late 19th and early 20th centuries. Syphilis cases rose in the United States between 1900 and 1920, and it was around this time that newspapers began using the term "syphilis":
Indeed, discussion of the disease only took place in specialized books and in medical journals, not in publications that most people would have read. Viewed as a subject beyond the "boundaries of decency," syphilis was thought to be a disorder that affected only the immoral. In fact, the American press, yielding to the desires of the common people, was so unwilling to deal with the matter that the Reader's Guide to Periodical Literature did not include "syphilis" as a heading until 1907, and the New York Times Index avoided the term until late 1917.
This general trend should also account for a strong increase and then peak in the late 1910s/early 1920s.