Can postdocs get a funded position by direct contact?
I think that what has happened is that there are stories floating around, rumors with some kernel of truth, that this approach works. Sometimes I get emails from folks who are highly qualified in their area, but have no relation to our work. I think the model of what has "worked" is probably misunderstood. 

I think the only thing that has ever really worked is for the "applicant" to send out mass emails to everyone they can find that's even remotely related until they hit on a few people who do have a position open and who direct them as to how to apply through proper channels. Occasionally these folks do actually have good qualifications and do get a position. The story of this process gets mangled, and people begin to think that mass emailing really works rather than realizing that it's targeted applications through the appropriate channels that worked but were found by luck. 

I mostly don't respond to these kinds of emails unless the candidate is actually highly qualified, related, and I just don't have anything for them. I got one from someone in the UK this week who is working in my field, looking for a PhD position, and already has 3 (!!) master's degrees. Unfortunately, we run an HPC center and don't have faculty positions or advising rights, so I wrote the guy back and let him know that. Rarely do these emails deserve such a response since the candidates are applying randomly to every name they can find, i.e. wet lab biologists applying to an HPC center.