You forgot to register the event 'register' => 'menu:title'  in the elgg-plugin.php.

'events' => [
    'register' => [
        'menu:title' => [
             '\MyPlugin\Profile\Menus\Title::register' => [],
        ],
    ],
],

Now create a file \mod\my_plugin\classes\MyPlugin\Profile\Menus\Title.php with this code:

<?php

namespace MyPlugin\Profile\Menus;

class Title {

   public static function register(\Elgg\Event $event) {

        $user = $event->getEntityParam();
        if (!$user instanceof \ElggUser || !$user->canEdit() || !elgg_in_context('profile'))  {
        return;
       }

       $return = $event->getValue();

       $return[] = \ElggMenuItem::factory([
               'name' => 'avatar:edit',
               'icon' => 'image',
               'text' => elgg_echo('avatar:edit'),
               'href' => elgg_generate_entity_url($user, 'edit', 'avatar'),
               'link_class' => ['elgg-button', 'elgg-button-action'],
       ]);

       return $return;
   }
}

Fantastic!

Thank you so much for the fast and easily understandable answer.

At least I had the folder path correct in the first try.

The idea for the menu was also the right one, but not fully correct.

And once again, I forgot to register the event in my plugin. That's a lesson you taught me before, but I forgot as I didn't do any updates on my Elgg since last Spring.

I applied the code already and it looks perfect now. 

Once again, thank you very much for your help!