As Kaiser has pointed out, the meta_value => $todaysString is querying for events where the value for key event_date is equal to todays string. I.e. events happening today.
Also, from the fact that you’ve used strtotime suggests you are sorting by timestamp – which you would like interpreted (and sorted as) a number rather than string. So rather than ordering by meta_vaue, you want to order by meta_value_num. Try:
$args = array(
'post_type' => $post_type,
"$tax" => $tax_term->slug,
'post_status' => 'publish',
'posts_per_page' => 1,
'meta_key' => 'event_date',
'orderby' => 'meta_value_num',
'order' => 'ASC',
); // END $args