The diaper pattern is a well-know anti-pattern. But the idea that all exceptions are thus evil is a giant leap of logic that is hard to swallow.
Let's say you want to read a file. You may get an exception for malformed paths, permission errors, filesystem errors and so on. It's unlikely that I will recover from any of these errors during runtime. The only thing I really can do is log the error, inform the user and send a report.
The checked exceptions problem on Java is one of overall language and libraries design, not inherent on the checked exceptions idea.
I prefer to catch Exceptions that I know how I want to handle and handle them accordingly. For other exceptions, I prefer to let them fly. If and when they happen, I can then understand the scenario and implement the desired logic.