I'm not 100% sure on this one just yet.. Auto opening on focus is usually associated with a hover action, perhaps we need active opening with space/enter keyboard action. Requires a bit more looking into the ARIA spec.
Popups appearing on hover (tooltips) and popups appearing after click (…popups?) are distinct use-cases, currently we abuse PopupButtonWidget for both. For example, the FieldLayout help icon really should have a tooltip rather than popup (and in fact it does in PHP OOUI, but not in JS).