For the first part of your question, you might start with my 2017 answer to ...Why did Lincoln have seemingly no personal security and how did it compare to other world leaders at the time?, though that shouldn't preclude other, more focused answers.
I can’t really speak as to why the Secret Service still handles financial crimes, but I’ve done some work on the Secret Service as presidential bodyguards.
The Secret Service conducted some investigations at the White House in the 1880s, but they first took on presidential security in 1894 when President Garfield’s wife (Frances) asked Secret Service Director William Hazen to assign detectives to the President’s summer home as a favor. Hazen agreed, despite not having the federal authority to use funds for this, so the President and the Secret Service kept this arrangement secret. Hazen continued the practice with President McKinley, until a Treasury investigation uncovered this policy. Hazen was demoted down to a field agent for misappropriating federal funds to provide presidential protection.
John Wilkie succeeded Hazen as director, but he also disregarded the Secret Service’s lack of authority in presidential protection. McKinley’s 1901 assassination exposed the Secret Service, but Wilkie proved to be an adept politician. Wilkie leveraged statements from prominent witnesses at the assassination and convinced Treasury Secretary Lyman Gage that his agents (and anyone else for that matter) couldn’t have prevented the assassination due to the ingenious way the assassin concealed his gun, and that the Secret Service would still be the most capable security for presidents.
In 1902, Theodore Roosevelt begrudgingly accepted Secret Service protection and Congress, which had originally resisted approving the Secret Service’s protection role, finally supported the agency’s handling of presidential protection, albeit informally. The Secret Service has maintained that role since then.
Gary J. Byrnes’s ‘Secrets of the Secret Service’ is a great source on all things Secret Service, as is Philip H. Melanson’s ‘The Secret Service: The Hidden History of an Enigmatic Agency.’
The personal correspondence of John Wilkie is also a great source, but at least as of 2015 you could only access it at the National Archives in MD. They’re long overdue for digitization.