So, since Garmin Connect seems to be down at least semi-permanently, is there any kind of offline software that will allow us to do the analysis and view the data?
The Garmin Training Software. You can import the FIT file from your watch. I use GTS to generate the file I can convert and import into my Polar SW, so that all my training files are in one place regardless of device.
There are other packages out there, but GTS is free.... it's not good enough to pay for to be honest.