The thing is all these letters prove ( and only MM can have leaked them) is that she wrote the letters , they don’t prove that the conversations took place , after all she herself doesn’t claim that she was actually there when the ‘conversations’ took place , they were reported to her by Harry, who can’t even remember if it was before or after they were married.
It’s a well rehearsed policy, what she did was create a paper trail around something which in reality never happened , or happened differently from how she has described it . But by putting it down on paper she has pushed the recipient into a position where they have to discuss the incident as if it really happened . She was laying a trap, it’s a bread crumb trail.