How do I retrieve the call-log content or mp3?

Are you trying to retrieve call recordings in Production Environment ?  If so please be aware that the Production Environment does not support accessing call recordings through API's.

If we are unable to use the API to download the recordings, what is the alternative?

function getURL($url, $login = false){
        $ch = curl_init ($url);
        
        if($login)
        {
                $poststring = 'LoginName='.$Username.
                              '&country=2'.
                              '&Password='.$Password;
                curl_setopt ($ch, CURLOPT_POST, 1);
                curl_setopt ($ch, CURLOPT_POSTFIELDS, $poststring);
        }
        
        curl_setopt($ch, CURLOPT_USERAGENT, "Mozilla/5.0 (Windows; U; Windows NT 5.0; en-US; rv:1.7.12) Gecko/20050915 Firefox/1.0.7");
        curl_setopt ($ch, CURLOPT_COOKIEJAR, 'RC_Cookie.txt');
        curl_setopt ($ch, CURLOPT_COOKIEFILE, 'RC_Cookie.txt');
        curl_setopt($ch, CURLOPT_RETURNTRANSFER, 1);
        curl_setopt($ch, CURLOPT_SSL_VERIFYHOST, 0);
        curl_setopt($ch, CURLOPT_SSL_VERIFYPEER, 0);
        $data = curl_exec ($ch);
        curl_close($ch);
        return $data;
}

$Username = '1713XXXXXXX';
$Password = 'XXX';

getURL('https://service.ringcentral.com/login/main.asp',true);

file_put_contents('data.mp3',getURL('https://service.ringcentral.com/mobile/media?cmd=downloadMessage&msgid='.$MessageID));

There's no need. I feel like the script I wrote is much better. If I were to use the API calls, I would have to throttle the amount of times I hit your servers while if I download each call recording using my script, I'm going to guess I can bypass the too many request API error and download each audio file one after the other.

You are actually using a method that we cannot guarantee will always work in the future.  I would encourage you to use the published API and wait for us to support it in production.

Since you are using an unsupported access method, it can be removed at any time.