Why don't we see the gas behind the black hole?
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.