BuddyPress.org

Try this to remove ALL activity feeds

function bpfr_hide_rss_feeds() {	
   remove_action( 'bp_actions', 'bp_activity_action_sitewide_feed' );
   remove_action( 'bp_actions', 'bp_activity_action_personal_feed' );
   remove_action( 'bp_actions', 'bp_activity_action_friends_feed' );
   remove_action( 'bp_actions', 'bp_activity_action_my_groups_feed' );
   remove_action( 'bp_actions', 'bp_activity_action_mentions_feed' );
   remove_action( 'bp_actions', 'bp_activity_action_favorites_feed' );
   remove_action( 'groups_action_group_feed', 'groups_action_group_feed' );    
}
add_action('init', 'bpfr_hide_rss_feeds');

This snippet will only show logged in users friends activities (on all activity pages of the site)

function bpfr_filtering_activity( $retval ) {
	$retval['scope'] = 'friends';		
		return $retval;
	}
add_filter( 'bp_after_has_activities_parse_args', 'bpfr_filtering_activity' );

Add to bp-custom.php or child-theme functions.php

More details about this on Codex

Both snippets are working. If really not for you, double-check that bp-custom.php is correctly using open/close php tags and if your theme doesn’t use some custom rss functions.

To remove the All members button :
remove line 19 in buddypress/activity/index.php from within your child theme.

<li class="selected" id="activity-all"><a href="<?php bp_activity_directory_permalink(); ?>" title="<?php esc_attr_e( 'The public activity for everyone on this site.', 'buddypress' ); ?>"><?php printf( __( 'All Members <span>%s</span>', 'buddypress' ), bp_get_total_member_count() ); ?></a></li>

and add class='selected' to the li containing friends item

If you remove the feeds display using the snippet, people won’t see anything even if they know the page url. And if this is not satisfying for you, you can write a rewrite rule to add to htaccess.