Google Analytics 4 - persistent values are not sent when configured using the 'set' command

Question

I want to configure an event parameter that will be sent automatically with every event I send to GA4. Including the page_view event and all other "Automatically collected events".

This is my setup configuration:

<script async src="https://www.googletagmanager.com/gtag/js?id=G-XXXXXXXX"></script>
<script>
    window.dataLayer = window.dataLayer || [];
    function gtag() { dataLayer.push(arguments); }
    
    gtag('js', new Date());

    // Doesn't work!!
    gtag('set', {
      'parameter_key': 'parameter_value'
    });

    gtag('config', 'G-XXXXXXXX');    
</script>

When I'm using the set command, the param_key event parameter is not sent with any event. It's totally ignored. However, when I move this object into the config command, it does work:

<script async src="https://www.googletagmanager.com/gtag/js?id=G-XXXXXXXX"></script>
<script>
    window.dataLayer = window.dataLayer || [];
    function gtag() { dataLayer.push(arguments); }
    
    gtag('js', new Date());

    // Works!
    gtag('config', 'G-XXXXXXXX', {
      'parameter_key': 'parameter_value'
    });
</script>

I want to use the set command because I have multiple properties, and I want this parameter to be used for all properties.

Anybody knows why it's not working and how to make it work?

Thanks!