A flashlight in space does propel itself, but that's not really a related question. A more related question is why does a flashlight in a fixed mount inside of a windowless spacecraft not propel the spacecraft (or, similarly, why does a fan inside of a sealed, ventless housing attached to an airplane wing not function as an engine.)

Of course, if there's some way this does work despite what we think we know about physics, its a pretty gigantic breakthrough (well, at least, understanding and practically applying it would be -- otherwise its just an anomaly that provides a pretty gigantic question mark...)

A flashlight in space would propel itself, since photons have momentum.

It does. But the propulsion is really, really tiny.