While I can see your point I don't entirely agree.
Yes, I think its vital for the good of everyone that the general population has a decent grasp of the basics of scientific principles. Unfortunately the attitude of all too many people in the general population is a gleeful ignorance where they proudly proclaim "I can't do mathematics", when they'd never be proud to say "I can't read!", to say nothing of the creationist crap that happens in the US where they proclaim "Someone has to stand up to these experts". As such I'd fully support any initiative to help raise the basic science teaching standards. However, I don't agree fully with the statement "If your work can't be explained to the average joe then you're not doing a good job".
Now part of this depends on what you mean by 'explain'. I can give a 2 minute summary of my PhD thesis concepts to anyone but it will utterly lack any detail. However, if they ask "But how did you arrive at your conclusion" then if I'm unable to go into details beyond high school level maths and physics then there is no answer I can give which isn't just "I can't tell you, you'll have to believe me" reworded. After all, a PhD is supposed to involve work sufficiently complicated it takes years to do and is something no one else has done before.
This sort of relates to 'catch 22' situations which occur on these forums when I'm 'discussing' things with cranks. A crank will ask "So how do you prove that then?" and I will give them a very superficial answer which outlines the methodology I'd use but doesn't give any details. This is because cranks invariably don't know the relevant details and thus I'm tailoring my reply to their understanding. However, they'll then say "That isn't proof, you haven't given any details!". I'll then give another reply where I get elbow deep in details and demonstrate the proof. Their response is "That's all maths, that doesn't count", in that they don't understand it so its not a valid answer.
Unfortunately the world is a complicated place and its a little naive to think that everything should be explainable to any level of specificity to Joe Public, some things are just inherently complicated. For instance, we all have a basic grasp of how the human body works but the intricacies and subtleties doctors have to deal with is something which the average person can't understand without years of training. That's why any specialised area of knowledge generally takes years of training, its outside of people's everyday experience.
If someone pushes for a 'detailed explanation' then unless they are sufficiently competent with the topic at hand to grasp the relevant methodologies then its often not possible to give an explanation of research to their satisfaction. This is not the fault of the researcher, it is related to the inherent complexity of much of the universe.