As a healthcare provider who has to was hands (I don't like hand sanitizer much) so often that my skin cracks and burns before half my day is over, I still either was my hands in front of the patient, or I ask if they mind if I use gloves (explaining my dry skin; otherwise they're likely to think I don't want to touch them because they are germy.) In doing so, I am only doing it for the patients' peace of mind. (Sometimes I use the same pair of gloves and wash them in the room.)
Trust me, the germs are everywhere, not only on my hands. They are on the chart, on the door handles, on the stethescope (I wipe my stethescope with an alcohol wipe if the patient is conscious and I have the time), the chairs in the waiting rooms, the table with the flimsy paper, the instruments, all over the patient sign-out area, etc. They are, in effect, unavoidable. And your child is not the only person who suffers. Every time a doctor who sees children moves to a new area of the country, they are repeatedly sick for a couple of years. We call it the "pedi-crud", and we get it from your kids who cough in our faces, touch their eyes/noses then the table, etc., etc.
That's me. What can you do?
In every patient room with a sink, there are handwashing instructions posted on the wall. Before you get started into any conversation, notice the instructions and ask the doctor if he/she believes handwashing is critical to fighting the spread of disease. They will probably answer yes, of course.
Well done, you've painted him/her into a corner.
Then answer his/her questions. If the doctor doesn't move to wash hs hands, just say, "Um, would you be so kind as to wash your hands? You've convinced me of it's importance." He/she should be quite happy to chuckle and do so.
If they do not, you may have a very pragmatic doctor, or one without empathy, but it's hard to tell. I would bring it to the attention of the office manager in this case.
I may have been brought up short by a patient (though not that I remember for hygiene); it's a humbling situation but not irreparable. Your relationship with the doctor matters.