However, when I add a featured image to the Blog page, it doesn’t show up.

When you assign a page to be used as Home page or as Latest Posts page, you are forfeiting that URL to be used in a different way than default. The Latest Post page does not use page templates either. WordPress configures the query for the intent, not the actual Page (well, Page for home is still a query for a single Page).
https://developer.www.ads-software.com/themes/basics/template-hierarchy/
The template hierarchy looks for certain theme files, in order, and uses the first one found. That theme file can query for a featured image, but it typically does not because of the nature of the intent of the page. Showing the latest posts can be on the home or on a separate page, so usually themes don’t look for a featured image.