I checked one of our sites on the mobile checker that you shared and was shocked to see a big fat ZERO.
Ya, that tool is very strict. It has to be because of the multitude of devices that may be used to view a document on the web. While it may work fine on your device, it may not work as well on another.
What's nice about the W3 mobile validator is that it will show you exactly what, why, and where there are challenges. It's a rather informative tool that will assist you in preparing your site for mobile users. Remember, much of that stuff that gets flagged can typically be dynamically changed (based on UA), revised, etc. to get as close as possible to that 100% compatibility score.
ZERO is not good. I find most sites to be in the 30-60% range. The two primary things to be concerned about are the functionality and the visual. Can a user complete the checkout process on their mobile device? Can they submit forms, interact with apps, etc?