Oops. I wasn't aware that you can change the display name also on the "edit profile" page. My suggestion was only for the user settings page.

To hide the display name input field on the edit profile page you will also have to override the view vendor/elgg/elgg/views/default/forms/profile/edit.php in your plugin. The change to be made is for the first input field and it would have to be changed into

if ($entity->isAdmin()) {
    echo elgg_view_field(array(
        '#type' => 'text',
        'name' => 'name',
        'value' => $entity->name,
        '#label' => elgg_echo('user:name:label'),
        'maxlength' => 50, // hard coded in /actions/profile/edit
    ));
} else {
    echo elgg_view_field(array(
        '#type' => 'hidden',
        'name' => 'name',
        'value' => $entity->name,
    ));
}

Additionally, with the Profile Manager plugin enabled, the change above will not have an effect on the edit profile page, because this view is already overriden. Instead the Profile Manager plugin uses the view mod/profile_manager/views/default/profile/edit/name.php. So, you would have to override this view also in your plugin with the modified view

<?php

if ($vars['entity']->isAdmin()) {
    $content = elgg_format_element('label', [], elgg_echo('user:name:label'));
    $content .= elgg_view('input/text', ['name' => 'name', 'value' => $vars['entity']->name]);

    echo elgg_format_element('div', [], $content);
} else {
    echo elgg_view('input/hidden', ['name' => 'name', 'value' => $vars['entity']->name]);
}

and place your plugin below the Profile Manager plugin in the plugin list.

P.S. no additional code in start.php necessary.

Thank you so very much! I think I got this now!

The plugin I have made from this I have added you as an author in the manifest... I would have not had a clue what to do without your help, and this has done a great deal to teach me how views work, and how dependencies relate. Thank you for all your help, and all you have taught me through this