The problem is exactly what you suspect, the query to search the posts is timing out. ACF uses WP_Query() to search the posts. When searching WP_Query() does a “LIKE” search on the words entered on both the content and the title of the posts.
This can be exacerbated if you have any plugins installed or code that modify the searches in any way and can be even worse if this code is trying to also include custom fields in the search.
