He´s not in charge. And even if he was, and even if he was the best ever at it(which the story suggest he really is NOT), that does not in any way translate to him ALSO being great at handling -this-. Also don´t forget that "in charge of" far too often means "only one big enough of a sucker to agree to do it" or "promoted beyond level of competence".
Specifically, Danny handles Personnel & Human Resources. And doesn't really have work for them. I mean, he's HR for a Dockworker's Association. Probably handles payroll processing and such.
As far as Danny goes, he handed over the papers and they were inconveniently processed before Piggot saw them. Now, given that it involves a minor, there are steps to removing them, yes. They involve a multitude of other things. It's never as easy to undo something you do with the government as it is to do it. You have to demonstrate why this is an unacceptable arrangement, obtain the consent of the parent(s) to undo what was done. In situations like this, where the parent's judgement could be suspect and thus, the minor child's well-being a potential compromised situation, other agencies may be brought in. It isn't as simple as Piggot saying 'We won't accept this', especially after the documents were processed to their appropriate locations and thus became binding.
As much as Piggot has authority over the PRT, she's still a cog in a government agency and does not have unilateral power. (Even leaving that aside, the affairs as they are give her more control over the situation than if she started that process of undoing what Danny did and cut Taylor loose. Then she has no control and there would probably be court proceedings to release Danny, plus the prospect of the YG sticking their nose in things to ensure that Taylor had a stable home life. (I kind of vilify them in Inclusion, but they can't be entirely terrible). And court proceedings... end up on a docket which is usually publicly accessible, so reporters assigned to check dockets for anything of interest to report on could find out about this whole mess... and then it's on the news.
Right now, she has some authority on the matter, as Danny made her a Ward, so she can keep those documents within her purview and thus, no one can see them such as busybody reporters, and Taylor isn't interested in rocking the boat and blowing the situation wide open, even if she is being unruly and violent, so Piggot can work a long game, find the right means to apply leverage and bring Taylor around. Danny could be a difficulty, but he's being rather shitfaced and she has people for dealing with difficult parents to keep him corralled. It isn't an ideal situation, but the alternatives could be worse.
