I am using the datepicker that comes with Silverstripe in an front-end form. I am also using the BootsForm Module. This is the code that works in Chrome:
This is not really a SilverStripe issue. You're just creating a HTML5 date-field and set the datepicker attributes. Most browsers don't support native date-pickers (yet), see caniuse.com
You could add some JS that progressively enhances your date-field with a date-picker though. Here's a tutorial that uses jQuery UI. If you're not using jQuery/jQueryUI, Pickaday is a good alternative.