Are ligand gated channels saturable?
Yes, they are saturable, because they have a finite number of 'binding sites', depending on the particular example you are looking at. Then, once they join the substrate to be transported, they need to release it at the other side of the membrane, which takes time (in very short time scales, but time nevertheless). Therefore, the saturation rate and speed of transport depends on the number of transporters on the cell's surface, the gradient differential, and the number of binding sites.

Facilitated diffusion, which needs a transporter as you point out, is a form of passive diffusion, which implies that there is no energy cost to the cell to transport the solute (as opposed to active transport). This is the key difference: energy. It does not mean that it goes as fast as osmosis or any other type of passive diffusion/transport, because, the transport that goes across the membrane (small gas molecules, and other small molecules) will clearly be faster. Also, passive diffusion does not mean that there is no upper limit to diffusion, in fact, diffusion is always limited by the concentration difference of the diffusive molecule across the membrane, no matter if this is passive or active transport. 

In short, it is simple/passive diffusion because it does not need energy. Whether is saturable or not is not what matters for this classification. In fact, all diffusion has a limit.