Replace "Week ... of ..." with new native date picker

[st3iny]

Is your feature request related to a problem? Please describe.

Clicking on the week select in the top left corner still opens the old date picker.

Describe the solution you'd like

It should use the new native date picker with improved a11y.

Describe alternatives you've considered

No response

Additional context

This requires some research if this is actually doable since it's a button. The native date picker usually requires an input.