Applying for postdoc grants

In our most recent "how can we help you?" thread, D writes:

I’m curious about the kinds of post doc funding made available by grant agencies like the early career funding available through the Humboldt Foundation, Marie Skłodowska-Curie Actions, Leverhulme Trust, and such. Would anybody be willing to provide guidance as to how one might go about obtaining such funding? I don’t mean just submitting the applications, which I take to be pretty straight forward (aside from the process of making one’s research appealing to the evaluators for such agencies). Some of these seem to grant postdoc funding for one’s individual projects at a host institution, where the host institution itself isn’t financially on the hook the person. How does one go about connecting with a desired host institution or faculty member at such a host institution? Is it all a matter of one’s antecedent connections? Is it ever acceptable to write a such a faculty member, in effect, saying “I’m interested in this project you’re working on, and it relates to my work like so. Would you be willing to sponsor my application to X?” Is there any other practical wisdom worth knowing about how to obtain such a postdoc? Perhaps this is obvious to job market candidates from Europe, but I'm a US grad, and this info seems harder to come by.

Euro responded:

I can say a bit about Marie Curie. They are very competitive, and more universities are spending money running workshops to help people write strong applications. You really need to have your research project connect tightly with a faculty member sponsor. So, clearly, it is best if you have already connected with the person, not only corresponding with them, but actually engaging with their research. I have sponsored two applicants (who did not get them ... which is not a surprise, given how competitive they are). But the evaluation reports explicitly talked about me, the sponsor, in the evaluation. (Incidentally, I was not the weak link). But they are not the sort of thing you just apply for. You need to have connected with your sponsor, and they need to support your project. As you note, universities love them because you essentially pay your own way. These sorts of grant schemes are common in Europe.

Euro then also added:

In most of northern-western Europe (Germany, Belgium, Netherlands, Scandinavia, Finland), the national funding bodies (and sometimes private foundations) support faculty research. One common form of support is to get money to support a project, which allows one to hire a post-doc or two (or more). The post-doc is tied to a very specific project, and is expected to work on it (though they can often spend some time on other research as well). These post docs are advertised like jobs (because they are jobs, and in many cases they pay well). But to be competitive you need to actually work on the topic of the post doc project. Here is where stretching things just does not work - I have seen applications that try. The most competitive candidates are already working on the topic, and have probably already read the supervisors work.

Any other readers with experience in these matters feel fit to weigh in?