Is it possible to transfer E-O157 from simply touching something with contaminated hands?
It's not unusual for a line cook to handle several things at once, but I can certainly understand your concerns. It would be preferable if the cook could pick up the burger with a seperate utensil, dedicated to the burgers. With someone else assembling the plate (bun, side dish and "fixin's").

The issue(s) with E-O157, as I understand them, are primarily a result of contamination during production. In NC, an establishment that grinds their own meat is allowed to serve the food product at any temp the consumer requests (even to the point of Steak TarTare - yum!). The reason is simply that they are in a more controlled environment and the risks are significantly lowered/eliminated. Businesses that have their ground beef processed off-site are limited to cooking it well enough to kill the bacteria. Typically, "well done".

Cross-contamination is a "thing", but a restaurant employee is more likely to think of it as: don't put the potato salad spoon into the spaghetti sauce, or don't stick the meat thermometer into the soup pot. Picking up lettuce and then reaching for a slice of tomato... is generally "ok". I do agree that it would be unappealing to see what you described. 

The US Dept of Agriculture/USDA provides much more information on their website.