I am not an expert but I think you are right and I misread it myself. After reading it again Gaussian elimination seems to be somewhat worse than O(n³) but other algorithms are as fast as fast matrix multiplication.
The details people are talking about in that MathOverflow thread concern the precision needed to represent the numbers that crop up during Gaussian elimination. (They get huge, and if you are counting bit or word operations, then Gaussian elimination is indeed terribly slow.) The post author is using a less detailed model in which he counts arithmetic operations. In this model, matrix multiplication and matrix inversion have the same asymptotic complexity.
