Is there a simpler version of WP Media?

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You can use ACF to do this.

  1. Set up your custom field for the user profile.
  2. Use the acf_form() to produce a form containing the field on your front end for the user to interact with.
  3. Customise your acf_form() arguments to use a “simple” uploader (this will not expose WP Media Library to users then)

Example: 

$field_arguments = array(
  'post_id' => 'user_'.$current_user->ID,
  'field_groups' => array(111), // Replace with ACF Group you created
  'form' => true,
  'uploader' => 'basic', // This is what hides the WP Media Library to the user
  'submit_value'    => 'Submit'
);