I can understand why Phillips, in October, and the Tories previously, did not agree to having a Home Office led investigation specifically into Oldham. Local authorities have primary responsibility for investigating local issues, the Home Office conducts national ones. The conversation has moved on, though, to whether a new national inquiry is needed.
I think its clear that the most immediately pressing issue is implementing the reccomendations of prior inquiries should be the most immediate concern, and I think it is right that should be the government's priority.
It isn't an either-or, though, you can implement reforms and have a new inquiry, too.
I've read a bunch of arguments from both sides of the current debate over the necessity/utility (or otherwise) of a new national inquiry. I don't give much heed to arguments from politicians or the likes of Musk, but I do think its fair to say that opinion varies even among people involved, who care deeply about the issue, are knowledgeable on it, and do not appear to have an alternative agenda.
We're told that most victims do not want another inquiry but I'm taking that with a pinch of salt, because I'm sure opinions vary among victims and I doubt they have been formally polled. I can see that some victims have spoken in defence of Jess Phillips in particular, so I don't think the OP's comments about Phillips are fair on that basis.
Starmer appears to have done a lot of good, on this particular topic, while he was DPP, hence the praise heapened on him by the Tories previously, before this became a political issue, recently.
The latest seems to be that Starmer will open a new inquiry only if victims groups call for it (although, he says, that in Jess Phillip's extensive engagement with victims, the overwhelming view has been that they so not want more inquiries).
I think making the decision based on what victims want is entirely reasonable, and I guess we wait to hear anything further from victims groups in that respect.
All to say, I think the OP is extremely unfair.
I do, also, think it is worth pointing out (as this is a thread about PMQs) that Badenoch has today admitted that she has not met with the victims, nor has she any plans to do so. I don't think it is right for her to demand a new inquiry and make this a party-political issue without at least taking the time to speak to any of the victims and take their views into consideration. It gives the impression that she doesn't care about the victims but wants to point-score.