You were exactly right! I took a closer look at the SD card, which I purchased pre-imaged from CrowdSupply, and it had only a single 5.7 GB partition (with no filesystem type specified). I tried to reformat it but the SD formatter threw an error (BTW, they are up to SD Formatter Version 5 now, and it does not allow you to enable "format size adjustment" anymore). There appears to be something wrong with the micro SD card itself. I'll contact Crowd Supply about swapping it out for a good one.
I took a known-good micro-SD card that I was using in a Raspberry Pi 3, reformatted it and put the "snickerdoodle_black_xenial_kinetic" image on it. This time the Windows PC saw three FAT partitions.
I put the newly-imaged card in the snickerdoodle and it booted right into Linux. It still did not show the white LED heartbeat during boot, so I guess that YouTube video I saw must have had a different firmware version. But I'm a happy camper now.
Thank you to everyone who helped me out with this problem!
- Chris