Have no experience of SW here, but I do know lots about the education, health and careers sectors where very often the biggest problem is getting those who need help the most to actually accept it.
My friend works in education with teenagers with extra needs - the kids who really need that extra help to get out and fulfill their potential. She spends long hours on the phone, arranging job placements for these kids - the kind of placements that many people would love their kids to be given - and so often, the kid doesn't turn up for work. My dad has had the same experience in the careers service.
Likewise, in health, it's often the most in need that won't take the services offered. So the NHS can offer screening services, walk in clinics etc and those in most need will choose not to attend.
My sister has worked in a project supporting women who have become sex workers to fund their drug habits. They are all entitled to state benefits, but will not engage with the process in any way, so they do not get the resources aimed at them. Even when the paperwork etc is set up for them, any further engagement required such as signing on, or attending meetings, falls by the wayside.
As others have said, we live in a world with finite resources. There comes a point where people have to help themselves. There are two sides to this, ie the poster who desperately wanted a speech therapy appt but had to wait for years. The resources aren't bottomless, and they should be targeted at those who are in most need, of course, but also those who want to engage and to use what's offered.