You can't liken giving a grant for a rape crisis service to medical research. This project funded a key worker to work with young women who have been raped and some of whom are suicidal. This kind of project doesn't come to an 'end'. By refusing to fund it further these women are left high and dry without their support and new victims can't even begin to access that support. In order to keep funding the service cuts have to be made elsewhere. I have experience of something similar personally and it is pretty inhumane to do this to the victims here.
Setting aside that much of the project funding was for outreach programme in schools, hence its relevance to CiN, let's deal with your point head on.
You make an absolutely compelling case for services of the type you describe to part of core provision funding from the public purse and I'd agree with you 100%. Sadly David Cameron's Big Society initiative was based on the premise that public funding could be reduced by offloading core public service provision onto the charitable/third sector. I've never agreed with this - charitable/third sector provision should extend but never replace core public services.
But once you enter the world where core public service provisions should be provided by the charitable/third sector, which has limited resources, you end up in a world where things we thing should be core public service provision are required to compete against each other for limited the charitable/third sector resource and not all can be funded.
So you might want the work proposed in the centre's application to be funded, and I wouldn't disagree. But that isn't the question - the question is whether you'd choose this application against another that you and I would also agree should be part of core public service provision. but where there isn't enough funding to support all the projects. So for example:
1. A project that supports young carers to allow them to have a childhood
2. A project that provides independence to severely disabled children
3. A project aimed at providing mental health support to the most severely children affected who may be considering suicide.
Or even:
4. A very similar project in another part of the country where this kind of support isn't embedded.
Now I'm glad I don't have to make key funding decisions between these types of application (mine are much more prosaic and less immediate), but I imagine the CiN funding panel members will be doing this all the time, and again sadly as you, I and no doubt they wish this wasn't the case and they could fund all if there were sufficient resources. But ultimately don't rail at CiN, rather get angry with governmental decisions to offload core service provision to the charitable/third sector which make these horrible decisions necessary.