The Childs, Kthari, and Rolando (2017) (CKS) algorithm can solve the quantum linear systems problem (QLSP) in $\operatorname{poly}(\log N, \log(1/\epsilon))$ time while the HHL algorithm solves it in $\operatorname{poly}(\log N, 1/\epsilon)$ time. So, if the CKS algorithm is superior, then why is the HHL more popular?
Additionally, are there any examples of the CKS algorithm being used to solve a system of equations on a real or simulated quantum computer?
HHL is more popular because it is much older (and it was the first in its category). The CKS algorithm (which you've been calling the Childs algorithm, even though the paper had three equally contributing authors in alphabetical order) is simply an improvement to the HHL algorithm in terms of its computational complexity.
The reason you haven't seen it being used to solve a real system of equations on a quantum computer, is because of the resources you'd need in order use the algorithm on even a matrix is that would be considered simple by classical computation standards.
