I'm afraid the author is correct in using big-O here instead of Omega or little-o.

For comparison, suppose that someone claims that for all x, sin(x) ≤ 1/2. That would just be wrong. If someone claims that sin(x) ≤ 2, then that is true, but not as informative as saying that sin(x) ≤ 1. There's something special about 1 here. If you want to highlight that, you can say that 1 is the least upper bound for sin(x). This is a bit of a mouthful, but we cannot achieve the same meaning by just replacing "≤" by "≥" or "=" or "<". Any of these replacements would just make the statement incorrect.

Big-O is something like the asymptotic version of ≤. For example, how many comparisons does heapsort need to sort an array of length n? If someone says O(n), that's just wrong. If someone says O(n^2), then that's correct, but not as informative as saying O(n log n). O(n log n) is special here, since it is the smallest complexity class that contains the number of comparisons. Again, it is a bit of a mouthful, but we cannot say the same thing by just replacing big-O by Omega, Theta, or little-o (the asymptotic versions of ≥, =, and <, respectively). For example, Theta(n log n) and Omega(n log n) are incorrect, since heapsort only requires a linear number of comparisons if the array already happens to be sorted.

The author argues that random memory access time is O(sqrt(n)), meaning that for large n, it might take up to constant * sqrt(n) time, but might also be faster if the memory to be accessed happens to be very close. Using Omega(sqrt(n)) instead would mean that random memory access can take an arbitrarily long time, but at least constant * sqrt(n). This is not what the author is trying to say.