The Java standard date API is so fundamentally broken that I have often considered simply adding Joda Time to the library extensions of the JVM so that it loaded on the classpath by default with the rest of the Java API.
If you have ever been tasked with retrofitting internationalization and time zones into a legacy Java application and have attempted to use the standard Java API alone, you will understand what I mean. I was able to turn thousands of asinine lines of code into less than a hundred. The productivity boost is inconceivable.
Further the standard Date API is not intuitive, where the fluid Joda API can be picked up in hours not weeks. Your analogy of trying to get to the island that is two miles away, is more akin to the following.
1) A motorboat that will get you to the island in only 6 minutes.
2) A makeshift bamboo raft tied together with vines during a hurricane steered by a crazy guy whose best friend is a volleyball.
As other answers have stated, the few drawbacks such as the ORM are even becoming non-existent as Hibernate now has plugins that allow for Joda type bean properties to be mapped to database date/time fields. JPA might also have an answer for this as well.
If your desire is for your application to have a minimalist footprint as a desktop application as it pertains to disk space then perhaps Java is not even the right language choice.