We have to override the template to remove the "Public" option. Please follow the steps below.

1. You will find the JS template from the BuddyBoss Platform on this path - buddyboss-platform/bp-templates/bp-nouveau/buddypress/common/js-templates/activity/parts/bp-activity-post-case-privacy.php

2. To override this template, you must copy that file into your child theme with this path -buddyboss-theme-child/buddypress/common/js-templates/activity/parts/bp-activity-post-case-privacy.php.

3. Copy the code below to your bp-activity-post-case-privacy.php or copy the attached file to avoid confusion.

<script type="text/html" id="tmpl-activity-post-case-privacy">
<div id="bp-activity-privacy-point" class="{{data.privacy}}" data-bp-tooltip-pos="up" data-bp-tooltip="<?php esc_html_e( 'Set by album privacy', 'buddyboss' ); ?>">
<span class="privacy-point-icon"></span>
<span class="bp-activity-privacy-status">
<# if ( data.privacy === 'loggedin' ) { #> <?php esc_html_e( 'All Members', 'buddyboss' ); ?> <# } else if ( data.privacy === 'friends' ) { #> <?php esc_html_e( 'My Connections', 'buddyboss' ); ?> <# } else if ( data.privacy === 'onlyme' ) { #> <?php esc_html_e( 'Only Me', 'buddyboss' ); ?> <# } else { #> <?php esc_html_e( 'Group', 'buddyboss' ); ?> <# } #>
</span>
<i class="bb-icon-f bb-icon-caret-down"></i>
</div>
</script>

4. Kindly make sure to add the codes we have provided before on your Child theme functions.php:
/*****************************
* Remove Public view on the Activity page
******************************/

function custom_login_css() {

    echo '<style type="text/css">
#loginform input[type=email],
#loginform input[type=text], 
#loginform input[type=password] {color: #000000!important;}</style>';

}

add_action('login_head', 'custom_login_css');

#activity-post-form-privacy label[for="public"] {
   display: none;
}

Note: Any modifications are considered as custom work already. Know more about our Support policy here: https://www.buddyboss.com/support-policy/