Short answer is: Big Meta Queries are incredibly bad and can quickly lead to the MySQL-Server not responding anymore. You should try to rework the meta_query to not have so many clauses.
Wordpress Post Metadata is stored incredibly inefficient when it comes to querying. Best advise would be to never ever EVER store data that you would need to ...
I found an answer to my own question:
If I use the tax_query arg then I could specify 'operator' => 'NOT IN' like so:
$events = tribe_get_events([
'tax_query' => array(
'taxonomy' => 'post_tag',
'field' => 'name',
'terms' => $current_tag,
'operator' => 'NOT IN' // ...