FAT really is the only widely supported filesystem that is certain to be supported by any recent OS. NTFS is out due to the poor support on anything not Windows based. HFS only works on OSX. Nobody appears to actually use EXFAT and that has even worse support than NTFS. Ext3/4 and the other Linux filesystems are only decently supported on Linux. You can get third party modules which allow access to these various filesystems on a non native OS but that is an extra layer of complexity which just isn't warranted.
The only alternative is to do what Apple do on their iOS based devices and not use USB MSC to present it as a drive. Apple run a service on the device which lets you send/receive files from the internal media partition which is actually in HFS format. The downside is that you have to use a special application i.e. iTunes to manage it.
For the Kindle to allow the computer to access the media partition, it has to unmount it itself which is why the Kindle goes into the special USB drive charging only mode. Having the Kindle and your computer both access the media partition at same time would cause issues as FAT was never designed to have more than filesystem driver accessing it.