I nailed them on their description that the simplex algorithm was like Gaussian elimination for systems of linear inequalities.
No, The simplex algorithm is selecting elementary row operations to improve the value of the objective function.
Gaussian elimination also uses elementary row operations, but that is about the end of the direct connection. In particular, how Gaussian elimination selects elementary row operations is quite different from how simplex selects them.
For more detail, I explained the role of artificial variables where we just 'glue on' an identity matrix. So, since we just glue on an identity matrix, we never, directly in simplex, go through anything like Gaussian elimination to GET an identity matrix. Yes, at times, for numerical reasons, one should 'reinvert' the basis in simplex, and here Gaussian elimination, etc., might be used, but such 'reinversion' is not really simplex.
The authors mentioned Gaussian elimination when they should have mentioned elementary row operations. Next, what is special about simplex is improving the objective function, not working with linear inequalities. Indeed, simplex doesn't work with linear inequalities; instead, we just use 'slack' and 'surplus' variables to get a system of linear equalities (with non-negative variables) before simplex even gets the problem.
The authors were being sloppy. I called them out on it. It was appropriate that I do so.
If you believe that my remarks about simplex are in error, then by all means make your claims.
But the issue is not me; it's the book.