I do not collect swabs (or operate forensic vacuum cleaners) at crime scenes. I get called in as a team consultant to explain DNA results (particularly partial ones) to LE (who generally do not have a lot of background in genetics or biology).
I used to go to crime scenes more often than I do now, and forensics has gotten way more sophisticated with each passing year.
But the basic pattern, which is to carefully start with swabs of the victim's body, pause, more swabs after the body has been initially moved in preparation for removal from the scene, is still true. The bedding (in this case) and the blood in the bedding would be swabbed - but I would expect the bedding to be preserved altogether (we haven't seen any proof one way or another of that - it doesn't need to be the entire mattress).
I collect DNA swabs from living people, sometimes in relationship to a crime, often in an open-ended investigation. By this I mean, I am the person who explains what the swabs will be used for, encourages the donor to swab themselves, has the person put the swab in solution, close it and put it in a forensic bag of some kind. Getting living humans to donate DNA (for any purpose) is not a sure thing. It helps if the person in my role has something in common with the people being asked, and in mental hospitals, patients are often more cooperative with a woman handler. I can get reference populations to cooperate with swabbing. I am then part of analyzing the results. I take extensive genealogies, aided by prior DNA analysis (and by GEDMatch, academic testing of DNA - there's lots, and of course, public records if that's needed).
But I've never taken swabs under the direction of an investigator (FBI or otherwise) who wanted a wide band of DNA from a community (I've never heard of that happening in my region, either).
I also have brainstormed with LE about how best to obtain a POI's touch DNA, but LE is way better at the actual scenarios, I'm just there to point out potential difficulties.