IANAP so don't take my word for it, but I would assume the photons would have to be emitted EXACTLY tangentially at EXACTLY the photon sphere. Just a tiny bit earlier or later or with a tiny bit of radial momentum and the photon would eventually escape or fall into the black hole. And a real black hole is not a perfect Schwarzschild black hole, it will almost certainly have at least some charge or angular momentum, surrounding matter will change the metric somewhat. But I can not tell whether that makes it more or less likely for a photon to stay in orbit but I tend towards less likely because this seems to add more possibilities to randomly push the photon out of its orbit one way or another.
My understanding is that photons are uniquely unstable. By virtue of traveling at a constant (the speed of light), they lack the self-regulating orbital mechanics of matter (where going deeper into the gravity well speeds you up, and going farther away slows you down.)
Dealing with spherical cows, before accounting for drag, you can describe the possible space for valid orbits as a volume, and you could describe the bounds of the velocity vector via another volume, for all positions and velocities at those positions that describe - at least on paper - mathematically perfectly stable orbits.
Dealing with a photonic cow, before accounting for drag, you can describe the possible space for mathematically stable orbits as not a volume but an area - the surface of a sphere. The velocity vector has a very specific magnitude (c), and a very limited range of possible directions (exactly perpendicular to the normal of the sphere's surface). The shape you would use to describe the bounds of this velocity vector have 0 volume, 0 area. It only has a length.
Both cows can be further perturbed by drag, tidal effects, non-point gravity sources, etc. to further worsen the problem. Individual collisions with particles of space dust will push the spherical cow from one mathematically stable orbit to another, until eventually they add up enough to most likely deorbit it. Individual collisions with particles of space dust will instead take the photonic cow from one mathematically unstable orbit to another, with a decent chance of it ending up on an escape trajectory instead.
