I assume you mean 'Write access' field in the form.

make the pages in the pages plugin always "private"

You need to create an event in your custom plugin:

elgg-plugin.php

'events' => [
        'access:collections:write' => [
            'user' => [
                '\MyPlugin\Pages\Views::onlyAccessPrivate' => [],
            ],
        ],
],

MyPlugin\Pages\Views.php

<?php

namespace MyPlugin\Pages;


class Views {

    public static function onlyAccessPrivate(\Elgg\Event $event) {
        
        $input_params = $event->getParam('input_params');
        $return_value = $event->getValue();
        
        if (empty($input_params)) {
            return;
        }
        
        if (elgg_extract('entity_subtype', $input_params) !== 'page') {
            return;
        }
    
        if (elgg_extract('purpose', $input_params) !== 'write') {
            return;
        }
        
        unset($return_value[ACCESS_PUBLIC]);
        unset($return_value[3]);
        unset($return_value[ACCESS_LOGGED_IN]);
        
        return $return_value;
    }
}

remove the access dropdown list 

You need to override \mod\pages\views\default\forms\pages\edit.php in your custom plugin 

In \mod\my_plugin\views\default\forms\pages\edit.php edit this code:

switch ($name) {
        case 'access_id':
        case 'write_access_id':

On this:

switch ($name) {
        case 'access_id':

Thank you Nikolai.

It works well !

But the writing dropdown list remains visible and displays "private".

Is there a way to hide it?

(I clear the cache before each test I run.)

Forgot about saving...

Add a new event in your custom plugin:

elgg-plugin.php

'events' => [
        'fields' => [
            'object:page' => [
                \MyPlugin\Pages\FieldsHandler::class => ['priority' => 501],
            ],
        ],
],

Create a new file \mod\my_plugin\classes\MyPlugin\Pages\FieldsHandler.php

<?php

namespace MyPlugin\Pages;

class FieldsHandler {
    
    public function __invoke(\Elgg\Event $event): ?array {
        $result = $event->getValue();
        
        foreach ($result as $k => $field) {
            if (elgg_extract('name', $field) === 'write_access_id') {
                unset($result[$k]);
            }
        }
        
        $result = array_values($result);
        
        return $result;
    }
}

Note: your custom plugin must be placed under 'Pages' plugin on the /admin/plugins page.

Absolutely perfect !

Thanks again.