Cannot connect to the Docker daemon when using custom Docker image in managed AWS CodeBuild environment

I want to use a custom Docker image in a managed AWS CodeBuild environment. Previously, I had successfully used the provided image Amazon Linux 2023 x86_64 standard:5.0, which comes with pre-installed Docker. However, using this or any other image as the base image seems like only a workaround and would either unnecessarily bloat my image or come with its own limitations.