It seems that the result of the "fromCallable" behaves a little bit different then an original Lambda function.

class A {
    private $name;
    public function __construct($name)
    {
        $this->name = $name;
    }
}

// test callable
function getName()
{
      return $this->name;
}
$bob = new A("Bob");

$cl1 = Closure::fromCallable("getName");
$cl1 = $cl1->bindTo($bob, 'A');

//This will retrieve: Uncaught Error: Cannot access private property A::$name 
$result = $cl1();
echo $result;

//But for a Lambda function
$cl2 = function() {
    return $this->name;
};
$cl2 = $cl2->bindTo($bob, 'A');
$result = $cl2();

// This will print Bob
echo $result;

Sadly, your comparison is incorrect.

// The equivalent to
$cl1 = Closure::fromCallable("getName");
$cl1 = $cl1->bindTo($bob, 'A');

// is most likely this
$cl2 = function() {
    return call_user_func_array("getName", func_get_args());
};
$cl2 = $cl2->bindTo($bob, 'A');

Executing one or the other Closure should result in the same access violation error you already postet.

----
A simple PHP 7.0 polyfill could look like this:
----

namespace YourPackage;

/**
 * Class Closure
 *
 * @see \Closure
 */
class Closure
{
    /**
     * @see \Closure::fromCallable()
     * @param callable $callable
     * @return \Closure
     */
    public static function fromCallable(callable $callable)
    {
        // In case we've got it native, let's use that native one!
        if(method_exists(\Closure::class, 'fromCallable')) {
            return \Closure::fromCallable($callable);
        }

        return function () use ($callable) {
            return call_user_func_array($callable, func_get_args());
        };
    }
}

I have two points:

It is possible to use Closure::fromCallable to convert private/protected methods to closures and use them outside the class.

Closure::fromCallable accepts late dynamic bindings using the keyword static if provided as a string. 

My code below demonstrate how a private static method can be used as a callback in a function outside the class.

<?php
function myCustomMapper ( Callable $callable, string $str ): string {
  return join(' ', array_map( $callable, explode(' ', $str) ) );
}

class MyClass {

    public static function mapUCFirst ( string $str ): string {
        $privateMethod = 'static::mapper';
        $mapper = Closure::fromCallable( $privateMethod );
        return myCustomMapper( $mapper, $str );
    }
    private static function mapper ( string $str ): string {
        return ucfirst( $str );
    }

}

echo MyClass::mapUCFirst('four little uncapitalized words');
// Prints: Four Little Uncapitalized Words
?>