Adressing the problem in the current correct answer "clicking the field shows the onscreen keyboard instead of the datepicker":
The problem is caused by the Browser behaving according to the type of input when clicking (=text). Therefore it is necessary to stop from focussing on the input element (blur) and then restart focus programmatically on the input element which was defined as type=date by JS in the first step. Keyboard displays in phonenumber-mode.
<input placeholder="Date" type="text" onfocus="this.type='date';
this.setAttribute('onfocus','');this.blur();this.focus();">