FSAA—full scene antialiasing—implies that the full scene is completely correctly sampled. In other words, even if little tiny slivers of triangles 1/1000th of a pixel wide cover the screen, each pixel's color will be correct as if you exactly correctly divided each pixel into the area covered by each little triangle and integrated the colors. Of course, this isn't practical, and it isn't how any hardware or software works. We approximate that result using a technique known as "point sampling." We evaluate, or sample, each pixel in one or more points, and assume that will be a good estimate.
As an aside, relative to your question, this is not an "inefficient" fake method—this is an extremely efficient fake method! It's a lot less work to do a small amount point sampling than actually evaluating the area across a potentially infinite number of points. Given that we're willing to sample more than one point per pixel, it's also efficient in hardware, and faster, to assume that you can sample these points (2, 4, or more) at the same time, and process them together. This is part of the cleverness of multisample antialiasing. In the long term, I hope that display resolutions will get large enough that the amount of sampling will become less important. In the short term, though, I believe that FSAA as we know it is here to stay.
Now, on to anisotropic filtering. You know, trilinear filtering was good enough for the best of the best flight simulators for many many years, and every time you or I have been in an airplane, the pilot flying it was probably trained in a simulator that used trilinear filtering! Anisotropic filtering is a relatively subtle effect of texture filtering, allowing severely oblique-angled textures to look both sharp and smooth at the same time. It's not really a "fake" method—it's a better estimate of the "perfect" filtering than either bilinear or trilinear filtering. And, there are still better techniques beyond anisotropic filtering, but they require even more effort and hardware.