Is there really no way to set a dynamic page, a shop page, or a blog page as homepage?
If, by a dynamic page, you mean a page whose content is determined by a dataset, then yes, you can make a dynamic page your home page. Here’s an example:
This home page is based on a record in a collection (‘Issue’) which provides the background image and the date information. The selected issue record also drives the repeaters that contain items from the ‘Stories’ collection and the ‘Artworks’ collection (those items are selected because they have reference fields that specify the selected issue).