$ rails db:create RAILS_ENV=production 　でエラー発生

やろうとしている事

Ruby on Rails　で作ったアプリをAWSでデプロイする過程で、DBはMySQLで、1つのEC2上にアプリと一緒にDBを作ろうとしています。

起きている問題

EC2上にUnicorn、Nginx、MySQLを入れました。
本番環境用のDBを作成しようとするとエラーが発生。

実際のエラー文

$ rails db:create RAILS_ENV=production

をすると、

Can't connect to local MySQL server through socket '/var/lib/mysql/mysql.sock' (2)
Couldn't create 'tempo_production' database. Please check your configuration.
rails aborted!
Mysql2::Error::ConnectionError: Can't connect to local MySQL server through socket '/var/lib/mysql/mysql.sock' (2)
/var/www/rails/tempo/bin/rails:9:in `<top (required)>'
/var/www/rails/tempo/bin/spring:15:in `<top (required)>'
bin/rails:3:in `load'
bin/rails:3:in `<main>'
Tasks: TOP => db:create
(See full trace by running task with --trace)

というエラー文が返ってきます。

試した事

MySQLの停止と起動。

$ sudo service mysqld stop
$ sudo service mysqld start

Redirecting to /bin/systemctl start mysqld.service
Job for mysqld.service failed because the control process exited with error code. See "systemctl status mysqld.service" and "journalctl -xe" for details.

というようなエラー文が返ってきました。
エラー文にある
$ systemctl status mysqld.service　を実行すると

   Loaded: loaded (/usr/lib/systemd/system/mysqld.service; enabled; vendor preset: disabled)
   Active: failed (Result: exit-code) since 火 2021-04-20 13:06:49 UTC; 28s ago
     Docs: man:mysqld(8)
           http://dev.mysql.com/doc/refman/en/using-systemd.html
  Process: 7921 ExecStart=/usr/sbin/mysqld $MYSQLD_OPTS (code=exited, status=1/FAILURE)
  Process: 7898 ExecStartPre=/usr/bin/mysqld_pre_systemd (code=exited, status=0/SUCCESS)
 Main PID: 7921 (code=exited, status=1/FAILURE)
   Status: "Server startup in progress"
    Error: 13 (Permission denied)

と返ってきて、ここら辺のエラー文の意味がわかりません。。

/var/log/mysqld.logの中を見てみたところ下記が。

2021-04-20T13:06:49.543528Z 0 [Warning] [MY-010091] [Server] Can't create test file /var/lib/mysql/mysqld_tmp_file_case_insensitive_test.lower-test
2021-04-20T13:06:49.543541Z 0 [Warning] [MY-010159] [Server] Setting lower_case_table_names=2 because file system for /var/lib/mysql/ is case insensitive
2021-04-20T13:06:49.544252Z 0 [ERROR] [MY-010846] [Server] MYSQL_BIN_LOG::open_purge_index_file failed to open register file.
2021-04-20T13:06:49.544313Z 0 [ERROR] [MY-010817] [Server] MYSQL_BIN_LOG::open_index_file failed to sync the index file.
2021-04-20T13:06:49.544817Z 0 [ERROR] [MY-010119] [Server] Aborting
2021-04-20T13:06:49.544993Z 0 [System] [MY-010910] [Server] /usr/sbin/mysqld: Shutdown complete (mysqld 8.0.23)  MySQL Community Server - GPL.
mysqld: File './binlog.index' not found (OS errno 13 - Permission denied)
2021-04-21T07:06:35.001596Z 0 [System] [MY-010116] [Server] /usr/sbin/mysqld (mysqld 8.0.24) starting as process 14680
2021-04-21T07:06:35.036077Z 0 [ERROR] [MY-010119] [Server] Aborting
2021-04-21T07:06:35.046084Z 0 [System] [MY-010910] [Server] /usr/sbin/mysqld: Shutdown complete (mysqld 8.0.24)  MySQL Community Server - GPL.

実行環境

database.yml

default: &default
  adapter: mysql2
  pool: <%= ENV.fetch("RAILS_MAX_THREADS") { 5 } %>
  timeout: 5000
  encording: utf8
  reconnect: false
  username: root
  password:

development:
  <<: *default
  database: tempo_development
  host: localhost
  socket: /tmp/mysql.sock


# Warning: The database defined as "test" will be erased and
# re-generated from your development database when you run "rake".
# Do not set this db to the same as development or production.
test:
  <<: *default
  database: tempo_test
  socket: /tmp/mysql.sock

production:
  <<: *default
  database: tempo_production
  password:

バージョン
Ruby 2.6.2
Rails 5.2.5
MySQL2 0.5.3

です。
初学者のため、拙い質問の仕方で申し訳ありませんが、何かわかる方いらっしゃいましたら、よろしくお願いします。