The importance of the full-row-rank assumption for the simplex method

Consider a linear programming model in the usual form ready for applying the simplex method. I understand that having the constraint equations' coefficient matrix $A$ be of full row rank means not having any redundancy in the constraints.

However, will $A$ not being of full row rank cause any problems in the application of the simplex algorithm? For example, if there is a degenerate basic feasible point that has not been preprocessed away, then the simplex algorithm (if unmodified) may go into an infinite loop and never terminate. How about when $A$ is not of full row rank?

$$P = \{x\in\mathbb{R}^2| Ax = b, x\ge0 \}$$, where $$A = \left[\matrix{1&1 \\ 2&2}\right]$$ and $$b = \left[\matrix{1 \\ 2} \right]$$.
Let the function we want to minimize be $$c^Tx$$, where $$c = \left[\matrix{1 \\ 0}\right]$$.
It is clear that $$x = \left[\matrix{0 \\ 1}\right]$$ is the only optimal solution to the problem. However this $$x$$ is not a basic feasible solution and, since the simplex method iterates over basic feasible solutions, the simplex method wouldn't succeed in finding the answer to the problem.
Further observations: Actually, the simplex method does not make sense if the rank of A does not equal the number of constrictions ($$m$$), because the method relies heavily on that hypothesis. Also note that the definition of basic feasible solution (http://www.math.chalmers.se/Math/Grundutb/CTH/tma947/1617/lectures/lecture9.pdf, page 4/30) is given in a standard polyhedron with rank $$A = m$$. Otherwise (if rank $$A < m$$), the definition is useless, for there is no subset of $$m$$ linearly independent columns of A, and therefore there is no basic feasible solution.