Ah, okay. That's called setting a STATIC front page for your site, by the way. From that, yes, you can move the 'posts' page.
Doing this adds that posts-page in breadcrumbs as parent and the corresponding menu-item gets the parent/ancestor class
This is actually an incorrect understanding of page parent and inheritance. The breadcrumb trail has nothing to do, in this moment, with your post type or post-type parent, and everything to do with organization.
Now, remember that Custom Post-types are POST-TYPES, not posts. Posts have different rules, and CPTs are more like pages. You have have a page be a child of another page, but a post cannot be the child of another post. They're 'linear', for lack of a better term (date based?) rather than hierarchical like a page or a CPT. So while a post gets
yyyy/mm/postname, a CPT (like a page) only gets
postname. This feature is by design, and isn't likely to change any time soon. CPTs are meant to flat out, like a Wiki, with everything at the same level.
Now that we have that out of the way.
You cannot make a 'page' and use that for your archive, the same way you can for normal posts. So you cannot make a page called 'My-awesome-CPT' and have it be 'assigned' as the go-to archive page.
But you can rename CPT archives.
Look up the
'rewrite' => array( 'slug' => 'cool', 'with_front' => true ),
If you change the slug, like the example did, to 'cool', then that would be your archive page.
On breadcrumbs, which is a different issue....
A quick check on my site that uses CPTs heavily, and I see: Browsing: / Home / Questions / [post title] in my breadcrumbs when I look at a question. Which makes me think whatever you're using to generate breadcrumbs is the issue, not CPTs themselves (though it's very possible the theme I'm using fixed something).