I've wanted this for years, and it must be something at the root of the database structure that prevents this. Desktop user here.

As a work around, we use one of the extra data fields (OTHER) in the customer record that DOES transfer over. It feels completely ridiculous that the phone number fields don't transfer, but yet they have extra fields that do. We have modified the OTHER field to be a custom PHONE NUMBER field. The main issue with this is that now you have the customer phone number in the wrong field of the customer record, and it doesn't sync up with the regular phone number field unless you double enter it (which is what we do). Then we turn on the OTHER field (now labelled as PHONE NUMBER) in the template so it can be printed and seen on screen.