Fisher's exact test was designed for small cell problems; once the table is sufficiently dense, then a chi-square test is usually performed because the approximation is far better. 5 is rather arbitrary, but the intuition is correct. The chi-square tests without correction is a perfectly valid approximation as the sample sizes grow large and the large samples obviate the need for the correction. I do not think we should avoid using the chi-squared test without correction altogether is correct.