For instance while it is often not written explicitly, for most positions the candidate is expected to have taken personal contacts with one professor that may be interested, and set up a research project together.
This is not the case in the US. Depending on exactly how one interprets what you've written, it could even be illegal to do what you've described without explicitly stating this in the job announcement. Many postdoctoral jobs in the US are open to all areas of math; some are grant-funded and will have a more specific focus, but this will be clearly stated in the job advertisement.
What is true is that you are very unlikely to be hired for a postdoc (in the US, or I would assume anywhere else) if there isn't a professor in the department who strongly desires to hire you, and it is not so likely (though far from impossible) that this will happen because they are looking through a long list of applicants (any good postdoc will get hundreds) and just happen to look at yours. Your prospects will improve a lot if you (or your advisor on your behalf) make contact with professors who might be interested in your research.