Because, as you point out, unemployment benefits run out, and because only about a third of unemployed receive benefits, the unemployment figures have always been derived from a survey. No, it's not some random telephone survey...the country is divided up in regions, like regions in a state are put into groups (and some groups have only one area in them), and then individual areas are randomly selected with a chance proportional to size within its group. So place like NYC, LA, Chicago, DC, etc are the only regions in their group and will be selected with certainty, while Centre County, Mifflin County, Union County PA are probably in the same strata (with other counties probably) and only one will be selected. Then areas within are randomly selected down to a set of individual addresses. Houses are in the survey 4 months, out for 8, back in for 4 (and that's regardless of whether the house changes occupants or not. Initial interviews and re-entry interviews are in person, and then phone option afterwards.
Sorry for the overly long explanation, I've just found that many people equate "survey" with "completely inaccurate telephone survey.
Back to the point...those in the house age 15 and older are asked if they worked the previous week and if not, then what had they done to look for work in the previous 4 weeks. Just reading the classifieds doesn't count as active job search. Pretty much everything else does.
I know people still looking for a job after four and five years of being unable to find employment. They're still looking, but their unemployment ran out long ago for some, more recently for others, are they counted anymore or are they assumed to "not want to work" as you erroneously try to paint the current batches of chronically unemployed.
If they looked in the 4 weeks between surveys then they would be unemployed, regardless of how long they've been looking. If they don't look that whole 4 weeks then they'd be Not in the Labor Force until they started looking again.
Oh, and I didn't erroneously say "not want to work" (and no one is assumed that they don't want to work)...I meant it. The majority of those Not in the Labor Force say they don't want to work...and that number and percentage has been growing.