Should Web games be playable on every device and platform?

I'll keep this short and sweet. Should a Web game (HTML5, JavaScript and CSS) be playable on every device and platform that supports the technologies it uses? What about if the experience will be ruined by physical device size or hardware capabilities?

Say I made the new WebGL and JavaScript version of Battlefield 3. Should that game be expected to run on mobile devices in the same way that it does on high-performance desktops? Even if your thumbs would cover the screen and ruin the gameplay? If not, how do you handle the situation on the mobile devices that support WebGL? Do you block them from viewing the game entirely and prompt them to play on a desktop computer? Or, do you instead provide a different game experience tailored to each device? For example with the WebGL Battlefield 3, perhaps the desktop version could see the rich first-person experience with the mobile instead having a 2D commander's overview of the game world that allows them to direct the desktop troops around the map. It's the same game but viewed in a different way depending on the device that you use to play.

I'm absolutely fascinated to hear what you think about this as it seems to fly in the face of the "write once, use everywhere" vision of the Web technologies that we're using to create games. Are games an exception to the rule? Perhaps we should see them differently to the way we look at websites, if only because we interact with games and websites in entirely different ways.