Washing your hands with soap and water is far preferable to hand sanitizer.
You are right as the water will wash away lumps of whatevers that hand sanitizer just moves around - even if it has sterilized it (to some degree)!
so yes, I agree with you 100%
Cooked food should be completely safe to eat. If you are going to 'pick up' anything it will be from the dish or utensils.
If you must eat out - are you talking street vendors? Why don't you carry your own small plate and utensils. They could possibly place the food directly on your plate without having to touch it.
ps - I don't think, if your hands air dry, you will get enough residue from sanitizer to affect anything.
Yes, if you are eating hand food, you should probably (preferably) wash or sanitize your hands before.
We need to be careful but don't get paranoid!
Meddle not in the affairs of dragons for you are crunchy and taste good with ketchup.