How can I exclude these fields from indexing?

All of those need to be explicitly enabled in Relevanssi settings, so just go through the indexing settings and disable everything. This will leave you titles and content. Then just enable tags from the indexing settings, reindex and you’ll be fine.

Those fields are always included in the query, but it doesn’t matter, because that content is simply not indexed and doesn’t exist in the Relevanssi index, so it doesn’t have any effect.

That’s not a surprising result: the default WP search is faster than Relevanssi, at least until it isn’t. The default search gets really slow when the site reaches certain size.

Since you’re already using Query Monitor, how long do the Relevanssi queries take to run? Are there are any particularly slow queries?

Are you using Relevanssi-generated excerpts? If you are, how much faster is the search if you disable the excerpts?

Which queries in the Query Monitor take the most time?

How big is your site, how many posts indexed by Relevanssi? Is the “throttle searches” setting enabled?

If you disable Relevanssi custom excerpts, does that help?

Ok, that settles it: with 44k posts, it’s just not possible to get faster results with multi-word queries from Relevanssi without the throttle. It’s just not technically possible, common words will fetch so many posts from the database that going through all them will take time, and there’s no way to make that faster. With rare words, the search may still be reasonably fast.

I assume if you look at the individual queries, if you have two search terms you have two queries, one for each word, and if the whole search takes about 8 seconds, each will then take about 4 seconds? That’s what’s happening there: each word you add to the search adds a new query, which may then add thousands of new posts to the search.

There’s a problem with the throttle and sorting posts by date, yes. Then again, if you’re not sorting by relevance and are mostly searching titles and content, you might just do well enough with the default WP search – it includes all the posts, sorts by date the same way Relevanssi does and is reliably faster.

Have you considered using Relevanssi Light? It’s much faster than Relevanssi or the default WP search, and doesn’t have a problem sorting posts by date.

What do you mean “limit the number of words”? There are plenty of filters that can be used to modify what is search and what gets indexed, so yes, it’s probably possible.

You can also try this:

add_filter( 'relevanssi_join', 'rlv_join_posts' );
function rlv_join_posts() {
    global $wpdb;
    return "LEFT JOIN ($wpdb->posts AS posts) ON (relevanssi.doc = posts.ID)";
}

add_filter( 'relevanssi_query_filter', 'rlv_order_by_date' );
function rlv_order_by_date( $query ) {
    return str_replace( 'ORDER BY tf DESC', 'ORDER BY posts.post_date DESC', $query );
}

This will sort the posts by post_date instead of relevancy, so you can use the throttle with the date sorting.