I'm a doctor, and this applies particularly to the US.
If a physician 'asked' me,
Look. I don't have time. I have other patients waiting. Do you want to start or not?
I'd say, "&#%$, no!", report them to the head of their department, and leave.
If you're looking for a polite way to get the person to bend to your will, just ask very nicely again. You can't do more. There are no magic words (with the possible exception of "Press Ganey") that will force your doctor to do anything.
Edited to reflect OP's edits:
Handwashing is better at removing pathogens from hands if it is done long enough and properly. However, that takes time (which is an increasingly precious commodity medicine today), and most physicians don't take the time to do it properly. In light of that fact, hand sanitizer is actually more effective than bad handwashing (with caveats.) But that's not what you're asking. You want to know how to ask a physician to wash their hands without offending them.
This is a fact: most doctors are happy to make their patients happy. If you feel uncomfortable, just speak up, be polite, and take "the blame"/responsibility for your feelings.
I'm sorry to ask, but I'm a bit of a germophobe. I know you used hand sanitizer, but would you mind terribly washing your hands? I'd feel so much better.
People can argue facts ('hand sanitizer is just as good'), but they can't argue feelings ('you feel fine with my use of sanitizer only'). If you make it clear that it's your feelings compelling you to ask, there really shouldn't be a problem.
If there is, I'd get the &#%$ outta there and report them to the head of their department.
Regardless of what's more effective, patient feelings matter. In the ER, I'd see what seemed like a ton of patients, sometimes the same ones several times before they left the ED. But I never touched a patient without washing my hands in front of them, which was not terribly important from a pathogen point of view, as I always washed after seeing a patient as well. But it was a matter of respect and consideration.
Sometimes my hands would get so raw that they'd crack and bleed. In that case, I would ask the patient if they minded if I skipped the handwashing and wore gloves instead, explaining why I didn't want to wash my hands (lest they think I thought they were too dirty to touch.) I never had a patient refuse that request.
My point is, a patient's feelings matter. If a physician is rude to you, unless it is an emergency, don't put up with it. If they don't care about your feelings, they may not care about other things either (like their quality of care.) I know some rude physicians who I would trust with my life, but they are few and far between.