That wouldn't have stopped them from writing they went into hiding for a century. It probably would've been better, because they could've written them as no contact for a century, and Georgiou would have been right to be more cautious. Burnham's story could've been a supposed Klingon attack with no evidence.
Except the Klingons were a mystery to the federation, you cant have that when everyone knows them intricately.
Cloaks wouldn't make a difference in the story.
Yes they would, since by that point of the late 24th century the Federation had a good deal of experience in dealing with cloaks. They'd be more worrying about the ships they can't see than the ones they can.