Most of the apps are little more than forms - there's a lot of logic in the way the different fields are filled out and the validation and business logic is probably quite complex. Since everything must be checked on the server anyway there's nothing stopping it from being migrated to pure HTML.
Is there a reason that Sungard HE can't handle the browser portion using JS or similar instead of JPI? No. But it would be a very slow migration process and, basically, all employees at many major universities would be stuck with older Java versions.
This is just one example that I'm (unfortunately) intimately familiar with. I'm sure there are many others that the typical HN crowd (small shops, latest technology, devops, etc) would never see.
Actually, that's already the reality of the Java ecosystem. Many libraries are tested to offer compatibility with 1.4, or even 1.3, and several flagship open source projects are not compatible yet with 1.7. With closed source and internally developed systems, the situation is even worse.
If I decide to use lambdas, I'll probably use Jython or Scala. I honestly don't expect to use Java 8 in the next 5 years.
The point is, if a Java developer starts working on a new product/project, it is very likely that he won't use Applets.
And one more thing, I know companies, that use Java based tools/apps, where they are not moving from Java 5 or 6. It's too expensive (installing newer JVMs, testing the environment and everything...) and they don't feel the need to do that.
Their main target is huge enterprise customers where Applets are still used today.
I'm really not seeing why they should care.