Data on the unemployment rate and the employment-population ratio come from the Bureau of Labor Statistics' "household survey," while the monthly jobs number comes from the more reliable "establishment survey." So what we have above is a chart where Kevin Drum, instead of looking at the employment-population number, just took the household survey jobs report and subtracted for population growth to get net job creation.
What you see looks an awful lot like a steady-state labor market with seasonality. We get more new jobs than new people most of the year, but fewer jobs than people in springtime. The only problem is that this is supposed to be a seasonally adjusted data series. I'm obviously not an expert in this, but I speculate that the seasonal adjustment algorithm has gotten miscalibrated somehow. One possibility is that aggregate population movement to the Sun Belt plus global warming mean that the cold weather disemployment effect of winter no longer matters as much as it used to.
To be clear, even if true this wouldn't alter the conventional wisdom about the basic economic situation. Whether the adjustment is right or wrong, we've settled into a high unemployment equilibrium with no sign of making up the lost ground anytime soon.