I can’t set meta_key in my custom post type query

That’s because ‘meta_key’ and ‘meta_value’ are not public query vars. In other words, you can’t use them in URLs directly, nor should you.

Instead, register a specific query var, like so:

function register_my_qv() {
  global $wp;
  $wp->add_query_var( 'my_qv' );
}
add_action( 'init', 'register_my_qv' );

Then, you can go to a URL like this: ?my_qv=foobar

All you need to do now is map your query var to the actual query you want to do:

function map_my_qv( $wp_query ) {
  if ( $meta_value = $wp_query->get( 'my_qv' ) ) {
    $wp_query->set( 'meta_key', 'some_meta_key' );
    $wp_query->set( 'meta_value', $meta_value );
  }
}
add_action( 'parse_query', 'map_my_qv' );