Button fields in Airtable are in fact formulas. The only way to show them in the form is after a record is created. Because of that, they can only be shown when a form is either editing a record or Save & Continue is enabled.
💡Tip: In the portal, you see records that already exist in Airtable, therefore the buttons are shown there. The same is not true when’s form is being used and no record exists yet in Airtable.
Script-Running Button Limitation
Buttons that run scripts will not work. This is because they are unavailable in the API and, therefore, cannot be supported by third parties outside of Airtable. But there's a workaround.