I'm a racer. I ride no hands to eat, change clothes, etc, often on training rides, and sometimes in races (not in the middle of the pack of course). It's very useful if you don't want to stop to do those things.

It's not so good for just riding. Sitting up like that increases your wind resistance and decreases your muscular effeciency- being bent over some lets you use your glutes effectively. Then there's the safety issue. An unanticipated side wind can cause you to swerve much more than if you had a hand on the bars. This is why I pick roads with wide shoulders or make sure there are no cars behind me when I am going to ride no hands to do something.

If you have elbow problems I suggest that you both get some PT to help fix it (possibly including a regime of weight work) and, if you don't think that it will get fully fixed, have your bike set up to accomodate it.

BTW, for some reason mountain bikes are harder to ride no hands than road bikes.