The title pretty much sums my question up. I've given a cursory glance to https://developers.google.com/youtube/iframe_api_reference and have searched online but all I have found are explanations on how to get an embedded video to autoplay (the first video only) while preventing the suggested videos list at the end. I think that perhaps the logic for playing suggested videos is only supported on youtube.com itself, a conclusion that might be supported by the autoplay button being outside of the video element.
As a worst case solution, I thought maybe if I could save a youtube.com page to my desktop I could modify the styling to only show the video. But alas, the page breaks when I try to load it from my hard drive as opposed to the website.
Edit: Take note that I'm talking about two different types of autoplay here. One is the autoplaying of suggested videos as seen in the picture, and that is the type of autoplay I want to be able to use in an embedded player. The other type of autoplay that I don't need help with is getting the embedded player to automatically play the video that is contained in its iframe src, aka the first video.