ujm:
You are correct in noting that there is no halachic obligation to pay alimony. There is about child support. Here, there is considerable misinformation. I am not a posek. So I will share what I gather from exact references.
Even Hoezer 71. This addresses the obligation to support one’s children. Rema says age 6, but clarifies this as whenever is the norm as per the chachomim of the times. While I have heard this being less than perfectly consistent among rabbonim, it is sometimes 18, other times 21. I have yet to encounter any rabbinically drafted agreement that specifies any age below 18.
You are correct, that support for children might be delivered in a manner that bypasses the mother. That could conceivably comply with halacha. However, there is a serious bump, and I say this based on hundreds of hours spent with dayanim, toanim, and frum attorneys. The court tosses agreements that do not pass muster in the court for adequate care/support for the children. Even with full agreement by the two parties, the court will force litigation if they feel the children are not being supported enough.
Not included in much of the dialogue here is that the father has several other expenses for the children, often serving as snags to the negotiation. Tuition is halachically the responsibility of the father. Does this include day care, tutors, etc.? Health care is not addressed in halacha. But it is a reality. Who takes the responsibility, and by what percentage? How do we insure the visitation occurs as agreed upon? As part of the civil divorce, the agreement gets filed in the court. Can the court be used to enforce it, or does this enter the discussion of mesira and arkaos? Is there basis in secular law to provide additional child support for the months of chagim?
In reality, toanim and batei din are keenly aware of their role vis a vis the secular court. And they tailor their work to avoid conflict. In many states, batei din have zero input into decisions about visitation and custody. In others, they have an opinion that is regarded, but not binding.
Lastly, no beis din has authority to enforce any of their rulings. In my experience, batei din look for the secular courts to back them up. And I am referring to the most “right wing”, frum batei din.
Smerel made a point that the videos of some agunah protests seem to involve onlookers who are uninterested in the case, but attracted to the drama. I have never witnessed these events, perhaps for the better. But I do hear the suggestion, and I do know of individuals who might well fit that profile.