Why don't we see the gas behind the black hole? In computer simulations of what a black hole might look like and many artist renderings of black holes, they all seem to show the brightly glowing gas BEHIND the black hole due to gravitational lensing.  In all such images (with a "horizonal" accretion disk) this light appears as a thin band on (the "top" or "bottom" of) the event horizon, that appears more and more intense as it gets closer to the event horizon.

But in the recently released image of the black hole in M87, no such lensing effect is visible.  Rather, it looks more like a hazy version of plain ol'Saturnian rings.  Why is that?

This link (http://rantonels.github.io/starless/) contains several images, and some math on what I was expecting to see.  This animated gif of a black hole from various angles, I think, best shows what I was expecting:
The reason is that your animated gif shows a geometrically thin, optically thick accretion disk, whereas the disk around the M87 black hole is a geometrically thick, but optically thin disk.

The main difference is that you do not see any evidence of the structure of the disk in the picture - it is nearly transparent. You do however see plenty of gravitational lensing - that is what the bright ring is. Many of our sightlines to hot plasma run through a small annulus at the photon sphere (at 1.5 times the Schwarzschild radius) making it bright. Because of lensing, the apparent size of this ring, as viewed from infinity, is enlarged to a radius of 2.6 times the Schwarzschild radius. Contrary to your question, we see photons emitted from all around the black hole in that bright ring.

There are many GRMHD (general relativistic magneto-hydrodynamic) simulations that do agree with what is observed.