I see one possible "exploit" (maybe we should rename this into "downsides"):
Often times, it is intentional and good for the game that messages have a limited lifetime. This creates conflict through differing memories. Easy facts remove that source of conflict.
Some people already save messages. Some even have set up scripts to do so easily.
That gives them something of an advantage in this area over people who don't have that kind of expertise or patience.
If everyone had access to a feature that would dump their message history, that would help a lot in leveling that playing field.
