A few years ago this changed. I now have an app on my phone that auto-downloads the newest episodes of my favourite podcasts in the background. Whenever I want to listen to a podcast, the app will have one available. Additionally, these apps allow me to speed up playback, read through show notes, and skip forward over ad reads or boring sections.
This is very convenient, and only began to exist in the last few years, hence the "new" popularity of podcasts in the last few years. I don't think we need to invoke in-car internet access to explain podcast popularity.