I used
binwalk to try to extract the filesystem of the Prime's firmware. In the directory I extracted to, I see a whole bunch of .zlib files, and nothing I try is able to decompress them. Is there any way to extract these files?
I don't think there is anything of interest to be extracted via binwalk. The majority of the firmware is just the raw executable. I think there is some limited data which is used to populate the initial filesystem, but nothing particularly interesting.
binwalk does extract from the zlib files it extracts, but the zlib stream in the foobar.zlib files seem to be fairly short and followed by a large block of data that isn't part of the zlib stream. If you feed one of the foobar.zlib files to the standard input of zlib-flate -uncompress, the standard output it produces is the same as the small foobar file extracted by binwalk. The bulk of the foobar.zlib file after the end of the initial zlib stream at the start of the file seems to be already in an uncompressed state, judging by the ASCII strings contained therein.
(On my Debian system, zlib-flate was installed as part of the qpdf package.)