Hi Jake,
I had Esri support try to figure this out with me, but my honest thoughts are Esri purposefully makes it difficult to embed onto a third party site because it's not Esri-owned. Support suggested I make a StoryMap page to embed on, but as a tourism organization where embedding a map on one of our landing pages would be the most visitor-friendly, unfortunately there are major flaws in the operation of the app. Not only that, but mobile traffic is the most popular traffic we have and the application refuses to configure to smaller screens although I have the app's screen change configuration sorted out. It works just fine as a StoryMap embed or on another Esri-based site, but not on any other third party platform.
It's worth noting I don't have coding experience, so this problem may be fixed with a bit of creativity and a couple lines of code, but I don't know where to start with that.
Kennedy