Surrogate-assisted search-based testing (SA-SBT) aims to reduce the computational time for testing compute-intensive systems. Surrogates enhance testing techniques by improving test case generation focusing the testing budget on the most critical portions of the input domain. In addition, they can serve as approximations of the system under test (SUT) to predict tests' results instead of executing the tests on compute-intensive SUTs. This article reflects on the existing SA-SBT techniques, particularly those applied to system-level testing and often facilitated using simulators or complex test beds. Our objective is to synthesize different heuristic algorithms and evaluation methods employed in existing SA-SBT techniques and present a comprehensive view of SA-SBT solutions. In addition, by critically reviewing our previous work on SA-SBT, we aim to identify the limitations in our proposed algorithms and evaluation methods and to propose potential improvements. We present a taxonomy that categorizes and contrasts existing SA-SBT solutions and highlights key research gaps. To identify the evaluation challenges, we conduct two replication studies of our past SA-SBT solutions: One study uses industrial advanced driver assistance system (ADAS) and the other relies on a Simulink model benchmark. We compare our results with those of the original studies and identify the difficulties in evaluating SA-SBT techniques, including the impact of different contextual factors on results generalization and the validity of our evaluation metrics. Based on our taxonomy and replication studies, we propose future research directions, including re-considerations in the current evaluation metrics used for SA-SBT solutions, utilizing surrogates for fault localization and repair in addition to testing, and creating frameworks for large-scale experiments by applying SA-SBT to multiple SUTs and simulators.