Toggle Button
Open the picker from a calendar icon next to the field. This page documents @pglejzer/datepicker-ui, the companion of timepicker-ui.
Calendar icon button
#ui.toggleButton takes a selector or an element. A click on it opens the picker, in addition to the input. The picker sets aria-expanded on the button, and its accessible name comes from your markup, so give it an aria-label.
index.html
<input id="date" /><button id="open-date" type="button" aria-label="Choose date">...</button>