Hello, and welcome to the discussion forum! You are welcome to ask questions here anytime. A good ear is important for any kind of musical learning. The ability to play with your fingers what you can hear in your head is a great skill. This frees you up from reading note by note, and is very helpful in memorization as well.

It's true that, at the extremes, there are two "camps" in the teaching community, one set teaching "classical" music written out in full, one set teaching "pop" music lead sheets, with just a melody line and chord charts printed. I like to think that neither approach is enough on its own, but that the most rounded musical instruction blends reading and improvisation (the Fabers call this holistic philosophy Analysis, Creativity, and Expression, or ACE).

The Piano Adventures series offers many opportunities to go "off book" and improvise or compose, or play by ear. Just look for the Creative! or Discovery! symbols for ideas.

As far as a second instrument goes, there are a number of things to think about. There is a lot to be gained from experiencing music-making in a variety of ways (e.g. breathing learned from singing or wind instruments, fluid arms learned from strings, etc.). At the same time, it can be a huge endeavor to manage practice time between two instruments. Are your children interested in playing another instrument? You don't want it to become a battle.

I hope that you are able to ask their teacher your questions. As a teacher, I would not be offended, but glad that a parent was interested and involved. Good luck!