BuddyPress.org

BP can’t do that out of the box.

You can extend it using custom post types.
No need to send them to wp-admin, they could create stories via a page within their profile.
Read and sort can be handled via the usual WP post functions, within or outside profiles.

So you know how to code a plugin that creates and handles cpts ?

Then something like this in the plugin class:

add_action( 'bp_setup_nav', array( $this, 'bp_stories_tab' ), 5 );

// create the tab
function bp_stories_tab() {
		
	bp_core_new_nav_item( array(
		'name' => __( 'Stories', 'bp-member-stories' ),
		'slug' => 'stories',
		'position' => 200,
		'screen_function' => array( $this, 'bp_stories_screen' ),
		'default_subnav_slug' => 'stories'
	) );
}	

// load template when  tab is clicked
function bp_stories_screen() {
	add_action( 'bp_template_content', array( $this, 'bp_stories_profile_screen' ) );
	bp_core_load_template( apply_filters( 'bp_core_template_plugin', 'members/single/plugins' ) );
}

// show the stories screen
function bp_stories_profile_screen() { 
	// etc. 
}