Upcoming Event: How do I sort database by custom date field, but ignore past dates?

You’re pretty close, but I wouldn’t be distracted by the ‘scope’ meta key. The following is untested.

<?php
$args = array(
    'post__not_in' => array($donotrepeat),
    "post_type" => 'event',
    'meta_key' => '_event_start_date', // name of custom field
    'orderby' => 'meta_value',
    "order" => 'ASC',
    "posts_per_page" => 1,
    'meta_query' => array(
        array(
            'key' => '_event_start_date',
            'value' => date('Y-m-d'),
            'compare' => '>=',
            'type' => 'DATE'
        )
    ),
);
?>

The meta_key is only present for ordering – meta_query controls the actual selection.

Note that I left your posts_per_page as is, even though it will limit the selection to one record.