Alexey

@vmarko, Does the usage of wp_cache_get/set in my theme, improves the object cacheing speed?

Fixed! I found the commit which is fixing the bug.
Now I can see the output of object caching.
One question: Does the usage of wp_cache_get/set in my theme, improves the object cacheing speed?

The default wp search is using Get method and not Post. I thought to edit wordpress core files but I really want to solve without rewrite functions.
So rest prepare its not a solution in this case

No , is not working
I tried to edit my code as this:

function rest_prepare_post_tag( $response, $object ) {
error_log( print_r( $response, true ));
    return $response;
}
add_filter( 'rest_prepare_post_tag', 'rest_prepare_post_tag', 10, 2 );

I suppose that the result from ajax-tag-search , is not passed via rest_prepare

Can I see the result on wordpress debug log file?

I know what is doing vardump. The problem is that when I tried, on Ajax page I see same result.
I tried by return var dump of data