Oswald, Proto, & Sgroi (2005). Happiness and Productivity. Journal of Labor Economics. Caveats below.
Mark's point about reverse causality (productivity drives happiness) is an important one, so please note: This was a randomized experiment, so more productive people couldn't self-select into the "happier" group.
Caveats: This is for narrow, synthetic definitions of happiness (watching comedy) and productivity (piece work)! It's fair to question the application to real-world knowledge work. I think the best line of skepticism is what it means to "create happiness." Showing funny videos at the start of the workday is very different from keeping people underutilized.
Why I think the guidance is still good: The podcast referring to giving directs enough work that it can't all get done. Not doing so risks leaving people feeling underutilized. "Producing happiness" is not keeping people underutilized so they don't feel too busy. It's also not avoiding short-term conflict for long-term benefit.
I do have some questions about the cognitive load generated by directs feeling that tasks are falling off the plate, but as long as everyone's proritizing effectively anyway and as long as everyone knows they're being evaluated on P1 and P2, not P50, then the negative effects should be minimal.
But it's more accurate to say "Giving directs what they think makes them happy doesn't drive productivity," not "Happiness doesn't drive productivity." Because there's some evidence it does. Just my egghead academic opinion....