While I agree, I think that if that's all then it's weak storytelling. She doesn't need blackmail, because she could have him arrested on the grounds that she is a witness to him having sex with minors. Her money and connections would bridge any gap in regards of proof. If she needs a blackmail that doesn't put him in jail, she could simply say she'll get him fired and without money (in her view) he can't provide for Ami.
It doesn't really fit that she would have Akira have sex with Touka/Tsukasa/Yasu for blackmail, or even amusement for that matter (because she overall seemed to want to manipulate things to a "better" outcome, like she's a matriarch fixing people's lives without them knowing). Her rationale of "let's have Touka have her heart broken in a 'safe' manner, by falling in love with someone I can control and therefore won't be harmful to the family", while bizarre, follows logic.
Her putting Tsukasa in danger on a whim does not. I think that's where most people started scratching their heads. You can argue that Touka has enough maturity to choose for herself to pursue a sexual relationship (while still a crime), but Tsukasa absolutely does not. So what is this about? Is she trying to "help" Akira by putting him into a situation that he fears, and then proving he's able to control himself? Why would she care though, and why risk her daughter in doing so? Tsukasa may be "disposable", but even then it doesn't fit that Tsubasa wouldn't care about having one of her daughters sexually assaulted while being a kid (if anything for her family reputation of having to deal with a broken person). Also, I don't really buy into Tsukasa's view of her mother not caring about her - cause if she didn't, she wouldn't put up with Tsukasa as much as she does.
In fact, about Tsukasa specifically, I'm not saying her problems aren't real, they certainly are. But you can view it from the other side: Touka has to deal with all the pressure of being the heir (and being a woman when they wanted a man). Tsukasa has been freed from that. What she views as neglect is also a privilege. It's both things, and that's why it makes it even harder for Tsubasa to deal with it, since the one that has all the responsability thrust upon her (Touka) behaves as a sane person, while the one that doesn't have responsabilities behaves as a brat. You can see the gap between how the two of them view the same situation when Tsubasa has Akira in her office and Tsukasa yells at her. She calls off the meeting and seems honestly hurt by the situation, following it by going after her supposed "spare".
Tangent aside, I hope Tsubasa is being influenced by the tree or has another angle, because her actions since the Tsukasa thing stopped making sense for me, and for a lot of people I'd wager.
