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So I would strongly suggest such a mini-extension only. Now to the code review.

// CREATE TABLE query
$db->run("CREATE TABLE t (name varchar(255))");

// INSERT with getting insert id
$id = $db->run("INSERT INTO t VALUES (?)", $name)->lastInsertId();

// single cell result. useful with count(*) and such
$name = $db->run("SELECT name FROM t LIMIT 1")->fetchColumn();
$count = $db->run("SELECT count(*) FROM t")->fetchColumn();

// one-dimensional array with names
$names = $db->run("SELECT name FROM t LIMIT ?", 10)->fetchAll(PDO::FETCH_COLUMN);

// calling a stored procedure
$stmt = $db->run("call foo(?)", $name);
$results = $stmt->fetchAll();
$stmt->nextRowset(); // clean the connection state

Other examples can be added on demand.

Update.

On a second thought I think that to keep such an array with default settings would be a good idea. So it's better to make your class like

class MyPDO extends PDO
{

    public function __construct($dsn, $username = NULL, $password = NULL, $options = [])
    {
        $default_options = [
            PDO::ATTR_DEFAULT_FETCH_MODE => PDO::FETCH_ASSOC,
            PDO::ATTR_EMULATE_PREPARES => false,
            PDO::ATTR_ERRMODE => PDO::ERRMODE_EXCEPTION,
        ];
        $options = array_merge($default_options, $options);
        parent::__construct($dsn, $username, $password, $options);
    }
    public function run($sql, $args = NULL)
    {
        if (!$args)
        {
            return $this->query($sql);
        }
        $stmt = $this->prepare($sql);
        $stmt->execute($args);
        return $stmt;
    }
}

But for the rest - I honestly cannot think of any improvement else.

Yes, you'll have to supply the connection credentials manually, but that's actually a good thing - it will make your class reusable. Just include it into another project and instantiate with different credentials! Or you can use another connection in the same project, if required.

So I would strongly suggest such a mini-extension only. Now to the code review.

$db = new MyPDO("mysql:host=$host;dbname=$db;charset=$charset", $user, $pass);

# Table creation
$db->query("CREATE temporary TABLE pdowrapper (
            id int auto_increment primary key, name varchar(255))");

# Prepared statement multiple execution
$stmt = $db->prepare("INSERT INTO pdowrapper VALUES (NULL, ?)");
foreach (['Sam','Bob','Joe'] as $name)
{
    $stmt->execute([$name]);
}
var_dump($db->lastInsertId());

# Getting rows in a loop
$stmt = $db->run("SELECT * FROM pdowrapper");
while ($row = $stmt->fetch(PDO::FETCH_LAZY))
{
    echo $row['name'],",";
    echo $row->name,",";
    echo $row[1], PHP_EOL;
}

# Getting one row
$id  = 1;
$row = $db->run("SELECT * FROM pdowrapper WHERE id=?", [$id])->fetch();
var_export($row);

# Getting single field value
$name = $db->run("SELECT name FROM pdowrapper WHERE id=?", [$id])->fetchColumn();
$count = $db->run("SELECT count(*) FROM pdowrapper")->fetchColumn();
var_export($name, $count);

# Getting array of rows
$all = $db->run("SELECT name, id FROM pdowrapper")->fetchAll(PDO::FETCH_KEY_PAIR);
var_export($all);

# Update
$new = 'Sue';
$stmt = $db->run("UPDATE pdowrapper SET name=? WHERE id=?", [$new, $id]);
var_dump($stmt->rowCount());

The listing above is the actual working code that can be run as is.

// CREATE TABLE query
$db->query("CREATE TABLE t (name varchar(255))")->lastInsertId();

// getting insert id
$id = $db->query("INSERT INTO t VALUES (?)", $name)->lastInsertId();

// single cell result. useful with count(*) and such
$name = $db->query("SELECT name FROM t LIMIT 1")->fetchColumn();
$count = $db->query("SELECT count(*) FROM t")->fetchColumn();

// one-dimensional array with names
$names = $db->query("SELECT name FROM t LIMIT ?", 10)->fetchAll(PDO::FETCH_COLUMN);

// calling a stored procedure
$stmt = $db->query("call foo(?)", $name);
$results = $stmt->fetchAll();
$stmt->nextRowset(); // clean the connections state

// CREATE TABLE query
$db->run("CREATE TABLE t (name varchar(255))");

// INSERT with getting insert id
$id = $db->run("INSERT INTO t VALUES (?)", $name)->lastInsertId();

// single cell result. useful with count(*) and such
$name = $db->run("SELECT name FROM t LIMIT 1")->fetchColumn();
$count = $db->run("SELECT count(*) FROM t")->fetchColumn();

// one-dimensional array with names
$names = $db->run("SELECT name FROM t LIMIT ?", 10)->fetchAll(PDO::FETCH_COLUMN);

// calling a stored procedure
$stmt = $db->run("call foo(?)", $name);
$results = $stmt->fetchAll();
$stmt->nextRowset(); // clean the connection state

Curiously, I already wrote a review for your class quite a while ago, Your first database wrapper's childhood diseases - so common the mistakes everyone makes with their first PDO wrapper.

In short, your wrapper adds very little to the original PDO, at the same time severely limiting PDO's features. I would strongly recommend to get rid of this class, and stick to the original PDO with only one addition, as your query() method is indeed a good idea. So I would make it just

At the same time, this mini-wrapper of mine allows you to get the resulting data in dozens of different formats, simply by returning the PDOStatement.

Curiously, I already wrote a review for your class quite a while ago, Your first database wrapper's childhood diseases - so common the mistakes everyone make with their first PDO wrapper.

In short, your wrapper adds very little to the original PDO, at the same time severely limiting PDO's features. I would strongly recommend to get rid of this class, and stick to the original PDO with only one addition, as your query() method is indeed a good idea. So I would make it just

At the same time, PDO allows you to get the resulting data in dozens of different formats, simply by returning the PDOStatement.