Why index "index_events_on_project_id_kind_emitted_at" is not used?

why_index_not_used.sql

DROP TABLE IF EXISTS events;
CREATE table events (
  id serial PRIMARY KEY,
  project_id BIGINT NOT NULL,
  kind TEXT NOT NULL,
  emitted_at TIMESTAMP NOT NULL,
  created_at TIMESTAMP NOT NULL DEFAULT CURRENT_TIMESTAMP,
  updated_at TIMESTAMP NOT NULL DEFAULT CURRENT_TIMESTAMP
);

SELECT floor(random() * (30-1+1) + 1)::int; -- random number between 1 and 100
SELECT (array['u', 'v', 'w', 'x', 'y', 'z'])[floor(random() * 6 + 1)]; -- random kind
SELECT ('2022-' || floor(random() * (12-10+1) + 10)::int || '-' || floor(random() * (29-10+1) + 10)::int)::timestamp; -- random date

INSERT INTO events (project_id, kind, emitted_at)
   SELECT
    floor(random() * (100-1+1) + 1)::int,
    (array['u', 'v', 'w', 'x', 'y', 'z'])[floor(random() * 6 + 1)],
    ('2022-' || floor(random() * (12-10+1) + 10)::int || '-' || floor(random() * (29-10+1) + 10)::int)::timestamp
   FROM generate_series(1, 20000000) i;
   
-- Query similar to my app (we will use this query all the time)
EXPLAIN ANALYZE VERBOSE
  SELECT COUNT(*) FROM (
    SELECT 1 AS one FROM "events" WHERE events.project_id IN (SELECT GENERATE_SERIES(10,30) AS index) AND "events"."kind" IN ('x', 'y', 'w', 'v') ORDER BY emitted_at desc LIMIT 30) subquery_for_count;
-- 2s with a Parallel Seq Scan that remove 2 222 386 rows

DROP INDEX IF EXISTS index_events_on_emitted_at_project_id_kind;
-- Create index with the same query WHERE order
CREATE INDEX CONCURRENTLY index_events_on_project_id_kind_emitted_at ON events(project_id, kind, emitted_at);

EXPLAIN ANALYZE VERBOSE
  SELECT COUNT(*) FROM (
    SELECT 1 AS one FROM "events" WHERE events.project_id IN (SELECT GENERATE_SERIES(10,30) AS index) AND "events"."kind" IN ('x', 'y', 'w', 'v') ORDER BY emitted_at desc LIMIT 30) subquery_for_count;
-- 6s no usage of the index
-- Why the index is not used?

-- Put emitted at as first column in the index
CREATE INDEX CONCURRENTLY index_events_on_emitted_at_project_id_kind ON events(emitted_at, project_id, kind);

EXPLAIN ANALYZE VERBOSE
  SELECT COUNT(*) FROM (
    SELECT 1 AS one FROM "events" WHERE events.project_id IN (SELECT GENERATE_SERIES(10,30) AS index) AND "events"."kind" IN ('x', 'y', 'w', 'v') ORDER BY emitted_at desc LIMIT 30) subquery_for_count;
-- 500ms usage of the index
-- Index Only Scan Backward using index_events_on_emitted_at_project_id_kind on public.events  (cost=0.56..1544080.16 rows=13430667 width=16) (actual time=6.219..406.525 rows=156003 loops=1))