BuddyPress.org

I’m bit amazed by this

enabled direct registration of new members and disabled email confirmation

As far as I know, a user registering from the front-end (managed by BuddyPress) is first receiving an email containing a secret key he needs to paste into the BuddyPress activate page to validate their account. But maybe you are using a plugin or some custom code to disable this..

Otherwise, you can achieve what you describe adding the following code snippets to a bp-custom.php file.

/**
 * Adds activities generated when a user activates their account when a user
 * is created using the wp-admin/user-new screen Add new action.
 */
function automatically_generate_an_activation_activity( $user_id = 0 ) {
	// Adds a "became a registered user" activity.
	bp_core_new_user_activity( $user_id );

	// Fake a user log in.
	bp_update_user_last_activity( $user_id );
}
add_action( 'edit_user_created_user', 'automatically_generate_an_activation_activity' );

I see, you can try the following code:

/**
 * Adds activities generated when a user activates their account when a user
 * is created using the WP REST API.
 */
function rest_automatically_generate_an_activation_activity( $user ) {
	// Adds a "became a registered user" activity.
	bp_core_new_user_activity( $user->ID );

	// Fake a user log in.
	bp_update_user_last_activity( $user->ID );
}
add_action( 'rest_insert_user', 'rest_automatically_generate_an_activation_activity' );