Using field values in page.html.twig
If you want to see the result of your code in THEMENAME_preprocess_field() you have to put {{ promo_date }} in a field.html.twig.

But in your question you asked about the page.html.twig. To access the field value in this template, you can use the node object, which is preloaded in the page template. But only if there is a node in the url of your page. Otherwise this will return an empty string:

{{ node.field_date.value }}