How can I remove an entry from the blog front page?
The blog module builds this page in the function blog_page_last().

The query in blog_page_last() does have a node_access tag so you could use hook_query_TAG_alter() to add a condition but may be tricky to just target that one query on that page.

I think a better option would be to use hook_menu_alter() to change the callback for the blog menu item to a function in your own module. In this function you can just copy the code from blog_page_last() but add a additional condition for your new promoted field.