I've worked for social workers and I can vouch for the fact that the type of person you describe only applies to the odd one or two. The personality mix is as diverse as you'll find in any community. They are, as we say around here, 'a right lot'. They're ordinary, hard-working people who (for the most part) genuinely care about the people on their caseloads. I have seen social workers break down in tears when things have gone wrong for their clients, or when they've been unable to help. I've watched them battle with themselves over which client to withdraw from their caseload because they've officially got too many. Once, during a citywide power cut, my whole team went out after their day's work to distribute blankets and hot food to hundreds of vulnerable people in the area - this at less than an hour's notice.
The problem is that most people only see social work when things go wrong. There is so much that goes right, so many happy endings, but of course, the press is never interested in that.
You'll find that such jobs are advertised in the Guardian because it's the leading jobs national and is the first place that many professionals go to when they're looking for career moves. Employers know that, and that's why they advertise there.
I don't know whether you'd make a good social worker or not, because I don't really know you. But all sorts of people become social workers, so no matter what your personality there's probably a role there for you somewhere.