HTTP/2 supersedes HTTP/1.1 to tackle the performance challenges of the modern Web. A highly anticipated feature is Server Push, enabling servers to send data without explicit client requests, thus potentially saving time. Although guidelines on how to use Server Push emerged, measurements have shown that it can easily be used in a suboptimal way and hurt instead of improving performance. We thus tackle the question if the current Web can make better use of Server Push. First, we enable real-world websites to be replayed in a testbed to study the effects of different Server Push strategies. Using this, we next revisit proposed guidelines to grasp their performance impact. Finally, based on our results, we propose a novel strategy using an alternative server scheduler that enables to interleave resources. This improves the visual progress for some websites, with minor modifications to the deployment. Still, our results highlight the limits of Server Push: a deep understanding of web engineering is required to make optimal use of it, and not every site will benefit.