That's a different question. It's similar, but the situations aren't really the same. Developing a desktop app that runs on multiple OSes either involves using something like Gtk and probably having a noticeably non-native app on at least some systems or writing separate user interface code for every platform. I don't have much experience in web development(or desktop development for that matter), but given that all commonly-used browsers share at least a certain subset of HTML, CSS, and Javascript, is it really as hard as creating a desktop app that looks native on every major platform? Since there's no such thing as a native look for web apps, it's not really something you have to worry about, so you just have to deal with the differing HTML/CSS/Javascript implementations across browsers.
Besides, an organization is and far more likely to switch browsers than to switch OSes because changing browsers is a less dramatic change that is less disruptive to their old way of doing things.
I think it's a lot more similar than you realize. It's just that most web developers don't have the luxury of deploying to a homogenous environment, so they take it for granted that the site has to work in half a dozen browsers.
Or, turn it around: would you insist your intranet apps support IE 6, even if using IE 6 is forbidden by IT policy?
I would insist all applications to warn users IE6 is forbidden by IT policy, to record its usage and to fire upgrade requests for the IT services team.
Besides, an organization is and far more likely to switch browsers than to switch OSes because changing browsers is a less dramatic change that is less disruptive to their old way of doing things.