Purchase events sent via Measurement Protocol source is always "direct"

I am sending purchase events on the server-side via the Measurement Protocol like so:

        'client_id' => $_COOKIE['_ga'],
        'events' => array(
            'name' => 'purchase',
            'params' => array(
                'currency' => $currency,
                'transaction_id' => $referenceCode,
                'value' => $value
            )
        )

Everything seems to function properly, except for the fact that all of them display "direct" as their source. Given the significant influx of traffic from Google Ads, it's reasonable to assume that at least one of the conversions originated from an advertisement. Should the system automatically establish a connection between user tracking using the client_id variable, or do I need to take additional action, like manually configuring the campaign_details? If manual setup is required, it does seem peculiar that GA4 wouldn't handle this automatically by utilizing the client_id.

client_id per Analytics mania: www.analyticsmania.com/post/google-analytics-client-id/#find

Thank you for this. The solution is indeed to remove GA1.1.​ from the beginning of the ID and send that as the client_id ​value.