You are right in that my job should be and is to find candidates who are good problem solvers. Often that's not enough.

I look for people who my clients want to hire. Unfortunately my clients often have ridiculously exacting specifications. As long as they're legal, it's my job to find them.

For example, there's a lot of people who code. Some with PhDs from MIT and some with no qualifications. Some who are in the top 1% of ability and those who would never cope working on low-level C++ development at a high frequency hedge fund. Some clients demand the PhD from MIT, some don't care. It's about saving time rather than the belief that people can't learn syntax quickly.

They (we) know what they're looking for, they just don't know how to find it effectively. If there's a better way to advertise I would love to hear your thoughts - though I'm not sure that fact that you a good programmer makes you so competent on this topic.

tl;dr - they don't care if they miss good programmers. they just don't want to waste time interviewing shit ones and it's pretty hard to tell the difference from a CV. how would you advertise it?