How do I add a date picker that is rendered from most the browsers?
I had a similar issue just recently within a client's intranet application, after Firefox added HTML5 date field support. (FF is their corporate browser of choice.)

You can still use any JavaScript based datepicker that suits your needs. (E.g. jQuery UI's datepicker.)

In your JavaScript bind this datepicker to all input fields of type date and then change the type of the input fields to text. This prevents the browsers from using their own datepickers.

PS.: It's not exactly a Drupal specific question, but tomorrow when back in my office I might add a code example that works for Drupal 8.