IMO, it's simply a case that Starfleet isn't today's navy simply transplanted into the future. While it shares many similarities and traditions with today's navies, but it's ultimately a different organization. I think the term "Federation flagship" is a term/honorific unique to Starfleet--and maybe only of the 24th-Century Starfleet--signifying the primary ship chosen to represent the Federation in major affairs with other galactic governments. This wouldn't preclude other flagships in a more traditional sense of the term, but "Federation flagship" is probably just a fairly new Starfleet thing with no previous historical precedent...
Excellent point, Starfleet is not a current navy, they even say their not a military arm but rather a exploration branch. However, they, like it or not, are the only defense the Federation has. So I think that examining through the lens of a modern navy is fair. Considering that the current US Navy does provide an exploration and humanitarian service for the country. They don't explore as much but are limited to earth only for now.
I guess one question to ask is when Starfleet was created what traditions did they establish themselves on? NASA, Earth's Navies, Air forces?