MySQL PRIMARY Index Not Used – Troubleshooting Low Entry Counts

this query :

EXPLAIN SELECT cmd.cmd_date, liv.liv_nom
FROM cmd
    LEFT JOIN liv USE INDEX(PRIMARY) ON cmd.liv_id = liv.liv_id
WHERE 1
ORDER BY cmd_date DESC
LIMIT 0, 20;

gives me a different result when the table liv contains 5 or 6 rows.

6 rows :

1   SIMPLE  cmd     ALL                 134331  100 Using filesort
1   SIMPLE  l       eq_ref  PRIMARY PRIMARY 4   cmd.liv_id  1   100 

5 rows :

1   SIMPLE  cmd     ALL                 134331  100 Using temporary; Using filesort
1   SIMPLE  l       ALL PRIMARY             5   100 Using where; Using join buffer (Block Nested Loop)

I've read somewhere (can't find where anymore :/) that MySQL can choose to not use a primary if there is too few rows in a table (here the limit seems to be 5). Is there a way to disable this ? Because with the index the query takes 80ms, vs 4s without index :/

I'm using MySQL 5.7.9

Thanks

Here the 2 tables, simplified.

EDIT :

CREATE TABLE `liv` (
  `liv_id` int(10) unsigned NOT NULL AUTO_INCREMENT,
  `liv_nom` varchar(100) NOT NULL,
  `liv_denomination` varchar(100) NOT NULL,
  `liv_public` tinyint(1) unsigned NOT NULL,
  `liv_slug` varchar(20) NOT NULL,
  `liv_ordre` tinyint(2) unsigned NOT NULL DEFAULT '99',
  `liv_id_crm` int(11) unsigned DEFAULT NULL,
  PRIMARY KEY (`liv_id`)
) ENGINE=InnoDB AUTO_INCREMENT=11 DEFAULT CHARSET=utf8

create table cmd
(
    cmd_id int(10) unsigned auto_increment
        primary key,
    str_id int(10) unsigned null,
    con_id int(10) unsigned null,
    pri_id int(10) unsigned not null,
    liv_id int(10) unsigned null,
    ... 20+ more columns with data only ...
    cmd_transaction_id int(10) unsigned null,
    constraint fk_bout_commande_bout_structure1
        foreign key (str_id) references structure (str_id)
            on update cascade,
    constraint fk_crm_commande_crm_contact1
        foreign key (con_id) references contact (con_id)
            on update cascade,
    constraint fk_crm_commande_livre_campagne1
        foreign key (pri_id) references prix (pri_id)
            on update cascade,
    constraint fk_crm_commande_famille_livreur1
        foreign key (liv_id) references liv (liv_id)
            on update cascade,
);

create index fk_bout_commande_bout_structure1_idx
    on crm_commande (str_id);
create index fk_crm_commande_crm_contact1_idx
    on crm_commande (con_id);
create index fk_crm_commande_famille_livreur1_idx
    on crm_commande (liv_id);