Top búsquedas (últimos 30 días, no bots; aquí no guardamos bot flag, así que todo cuenta):

SELECT q_term, COUNT(*) AS n
FROM wp_search_log
WHERE ts >= NOW() - INTERVAL 30 DAY
GROUP BY q_term ORDER BY n DESC LIMIT 100;


Búsquedas sin resultados (qué falta en tu índice):

SELECT q_term, COUNT(*) AS n
FROM wp_search_log
WHERE zero_results = 1
GROUP BY q_term ORDER BY n DESC LIMIT 100;


CTR por consulta (clics / búsquedas):

SELECT s.q_term,
       COUNT(DISTINCT s.id) AS searches,
       COUNT(c.id) AS clicks,
       ROUND(COUNT(c.id)/COUNT(DISTINCT s.id), 3) AS ctr
FROM wp_search_log s
LEFT JOIN wp_search_click c ON c.search_id = s.id
GROUP BY s.q_term
HAVING searches >= 5
ORDER BY ctr DESC, searches DESC
LIMIT 100;


Posición media clicada por consulta (calidad percibida):

SELECT s.q_term,
       AVG(c.position) AS avg_pos,
       COUNT(*) AS clicks
FROM wp_search_click c
JOIN wp_search_log s ON s.id = c.search_id
GROUP BY s.q_term
HAVING clicks >= 5
ORDER BY avg_pos ASC;   -- más bajo = mejor


Embudo por dispositivo:

SELECT s.device,
       COUNT(DISTINCT s.id) AS searches,
       COUNT(c.id) AS clicks,
       ROUND(COUNT(c.id)/COUNT(DISTINCT s.id), 3) AS ctr
FROM wp_search_log s
LEFT JOIN wp_search_click c ON c.search_id = s.id
GROUP BY s.device;


Refinamientos (siguientes consultas en la misma sesión, 10 min):

SELECT a.q_term AS q1, b.q_term AS q2, COUNT(*) AS n
FROM wp_search_log a
JOIN wp_search_log b
  ON a.visitor_id = b.visitor_id
 AND b.ts BETWEEN a.ts AND a.ts + INTERVAL 10 MINUTE
 AND b.id > a.id
GROUP BY q1, q2
ORDER BY n DESC
LIMIT 100;