Aren't the reasons obvious? Firstly you always have relevant uniques and traits. Making the most of uniques and traits has been a mainstay of Civilization strategy since Civ3, and a key part of its replayability. Secondly, the start of a game, where everything is new, no one is too weak or too strong, and everything to play for; has always been the most fun part of Civ games. By having crisis (partially) wipe the slate clean and picking up with a new Civ we essentially get 3 early-games per game, and less tedious end-game. These are pretty huge mechanical reasons to have civ switching, which go to the heart of some fundamental problems
all previous iterations of the franchise have had.
By contrast, the cons seems to be limited to breaking
some people's immersion because the imagination required to tie it into a narrative (as in
#88) breaks their immersion by adding a slightly different kind of absurdity than has been present in past civ games. Immersion is not nothing, but I'll pick mechanics over a specific interpretation of fluff any day of the week.