I never used Processing on Linux, so the information I provide might be inacurate. I think the authors will clarify the issue better than me...
But, maybe I might address some concerns.
1) According to the GPL, the source code is available, and you are free to compile it with whatever tool you want. After all, that's a common practice for open source software on Unix platforms.
2) There is no installation, so no opportunity to show Java license at this step, as some other softwares do, but the LICENSE file is right in the java folder of the distribution.
3) There is a distribution without Java. It is a Windows distribution but I wouldn't be much surprised if it works on Linux with minimal changes.
Actually, the main difference I see is processing.exe instead of
processing shell script. Also the Windows dist has a video library not seen in the Linux one.
OK, actually it seems the jar files are slightly different. For example the Linux pde.jar isn't compressed, perhaps by mistake, and has a linux folder in the lib one, while the Windows jar has a windows folder.
Now, I am a supporter of a distribution without Java: it makes the download much smaller, which can make a difference if you try and follow all updates! Maybe we could have a mixed Java-less distribution, with both processing.exe and processing shell script, linux and windows folders (they are not so big), etc. unless there are major issues with this approach.
The MacOS dist is another beast, being packaged as a DMG file.