It is a perfectly normal and natural reaction to freeze up and be unable to think when you are in the moment of witnessing violence and aggression - this is part of fight or flight, because you are not at that moment in need of logic to evaluate the situation or work out what is best to do to protect somebody else long term, your priority is short term survival and usually what that means is to make yourself small and not do anything which would attract attention to yourself (it's really fight, flight or freeze - and freeze is typically the safest solution.)
It's good that you've been able to process it retrospectively and make decisions like contacting the school and documenting the injuries.
Don't judge your cousin too harshly. Of course you would protect your children from a stranger who did something like this out of the blue, but abuse victims are not in that situation, the abuser starts small and consistently ramps up so that it's never any one incident which partially means that it never really seems "that bad" because they are always having their perception of normal messed with and changed. Part of this is shame that they have put up with other forms of abuse and they feel that they can't leave "just" for this one, almost like they don't have the right. In addition, imagine that somebody told you - urged you - that your DH, the one you know and love and trust and rely on - that he was dangerous to your children and that you had to leave. Or even if one night he acted completely out of character and hurt them. Would you really find it easy to leave, or would it be so unbelievable that you'd look at other possibilities first - that he was perhaps ill, or had misjudged something, that the person was lying, that they were mistaken or mad? Particularly if he was repentant, apologetic and horrified and you believed he'd never do it again. The thought of having to instantly and totally reevaluate something which feels so stable in your life would be like a punch in the gut and your initial instinct would be to make it go away and excuse it or make like it had never happened. If you were faced with your DC's distress or some injury, then hopefully you'd bring yourself to act, but that wouldn't stop it being an utterly horrendous, distressing thing to contemplate that your relationship/marriage, which was perfectly fine just half an hour or a day before, was over. It's no different for an abuse victim. They don't see their marriage as any less valid or loving or supportive than yours, even if it seems that way from the outside. Usually an abuse victim feels exactly the same about their partner as the way anybody else does, they are not constantly living in fear and intimidation, they love and respect and have a life with that person.
It feels differently when you are out, and hopefully this is all a catalyst to change. You're also right that she may feel that the best way to protect her child is to be there, rather than split and risk unsupervised contact, as she may feel that she is capable of lessening or intervening in the abuse. You must understand this because you felt part of this yourself when earlier in the thread you mentioned anxiety about reporting in case the family shut you out and you lost contact with your nephew. Ultimately this is a false belief, because the person who believes they are "protecting" the child almost never has any influence over the abuser but it is a powerful belief all the same. If she is so ground down that she can't even begin to think about a plan to escape then this may combine with a feeling of hopelessness about the situation as well.