Took her ID and verified it was genuine with the issuing authority. Job done.
Indeed. Which is exactly what the Irish didn't do. They started a hare running of checking whether the ID was genuine but mis-issued, which is a whole other world of fraud.
It's trivially easy to check if a passport is genuine if you are the police of the issuing country: you quote the passport number, and you are shown the passport office's copy of the photograph. You might even be able to do this yourself for a passport with a chip in it with the right equipment: I don't know when Ireland started issuing them, but not all current UK passports have chips in.
But here the police went all Frederick Forsythe and, starting from the assumption that the passport was the result of a fraudulent issuing process, asked for evidence stronger than you need to produce to obtain a passport in the first place. And they did that without the slightest evidence, other than a racist tip-off on Facebook.
So here's what you think is "reasonable". A racist reports a family, anonymously, on Facebook, for having an abducted child. Not a particular abducted child, but just "oh, that doesn't look right". The police rock up and are shown a passport and some other ID (let's put aside that a lot of children in the UK don't have passports, or any other form of photo ID, particularly children in marginalised communities who don't have the twenty quid a year it costs to maintain a passport). The police immediately say that the passport is not sufficient, and start to perform a primary background check.
You still have answered why any of this requires removal of the child. The child was not in physical danger. There was no suggestion that the child was being abused, or, at least, abused to any extent which would of itself provide grounds for removal. We're always told that removal is incredibly hard, and children being beaten black and blue can't be removed because of that. But here, with no substantive abuse allegations, with valid ID documents, but merely on the say so of a racist rant and the police suddenly deciding that passports can't be trusted, a child was removed.
My neighbours have adopted a child from Vietnam. Suppose I phone the police up and report them (your Mrs Bloggs example). Hospital records? Good luck with that. DNA testing? Obviously not. What evidence could they produce to prevent their child being removed for as long as it took to re-check the entire adoption process? And then, for fun, what happens if I report them again the following month? And the month after that? After all, they might be killing Chinese children and getting a fresh one shipped in each month, and we can't be too careful, can we?