The psychological peculiarity of a person is such that if a person is doing something and wants it to remain a secret that he is doing it, the realization that someone else knows it will unsettle. So Suguha doesn't need any proof, just an anonymous message saying, "I know you're cheating on Kirito with Inoda." Asuna can think all she wants about who sent it and how they found out, she can even think about Inoda, but it doesn't matter. The main thing is what the effect of this will be. You're right, Asuna will indeed return to Kirito at first, which contradicts Suguha's goals. But this will give Suguha time to investigate everything in more detail and possibly strengthen the blackmail, already supporting them with evidence. But Asuna has already fallen pretty deep, and how long will she play the good girl without quality sex?
For example, it can be one of the endings. Suguha collects evidence and tries to distance Kirito from Asuna as much as possible, then blackmails Asuna to peacefully break off relations with Kirito herself, otherwise she will post the recordings online. And he just takes the slightly broken Kirito for himself. And then, a little later, her tells that Asuna cheated on him and shows him, for example, videos or they meet pregnant Asuna dressed as a whore and Inoda, somewhere on the street. NTR's goal is achieved, Kirito realizes that Asuna cheated on him all the time with the fat bastard. He essentially lost compared to Inoda. Profit.
In fact, it doesn't matter who does it or for what purpose, it can just lead to a lot of interesting scenarios. And that would have pushed Suguha's plotline and possibly brought some chaos to Asuna's plot. She's like a log right now, just going with the flow.
