When I use the datalist element with a text input, Chrome shows an arrow on the right and the list drops down when you focus the input. Firefox, however, shows neither until you start typing, at which point it suggests just those items that match your input. So there's no way to see the full list of options.

I think Chrome's behavior is correct here, but the larger point is that precisely because these are native elements, when they don't work there's nothing you can do. So your only option is to reimplement them from scratch.

Datepicker was fundamentally broken in many ways in safari for a long time. And it still doesn't have the functionality that most apps need, so it's pretty pointless.