Thanks guys; it ended up being a completely separate issue. I'll document what I did below in case someone has a similar problem in the future.
I thought maybe Hekate refusing to toggle AutoRCM could be related to the problem, and I guessed Picofly was preventing AutoRCM from being toggled. After removing the Picofly, I was able to toggle AutoRCM, and I reinstalled the Picofly. This ultimately didn't do anything useful: instead of starting in RCM mode when plugging in a cable while powered off, it started the official bootloader. I turned AutoRCM back on, and I was back to where I started. So failing to toggle AutoRCM is a red herring.
(An aside: Hekate 6.1.1 incorrectly reported AutoRCM as being off, and I couldn't toggle from the off state. However, with Hekate 5.0.0, it correctly reported AutoRCM as being on, being unable to toggle from the on state. So you might not be able to trust what Hekate reports for AutoRCM when you have a Picofly installed.)
After that, I started checking for continuity between the eMMC connector and various parts of the board, using the Picofly install guide as a reference. I'm using a Picofly kit from Aliexpress, which contains a board that plugs into the eMMC connector and a flex cable to solder to the APU capacitors, so I didn't need to solder to the board besides at the APU capacitors. I was just checking for any signs of board / trace damage out of desperation.
I ended up finding what I believe to be two missing capacitors on my Erista board: the 3.3V point in the attached image, along with the capacitor north of it. (Image taken from the Picofly Install Guide). Not only were the capacitors missing, but the north pad of the 3.3V capacitor was bridged with both pads of the capacitor above it under a glob of solder. I got my Switch used years ago, and I've never had a problem with it prior to the Picofly install, but I guess a previous owner tried to mod it and got sloppy.
I wicked away excess solder and reflowed the pads to remove all the bridges, so that none of the pads are connected to each other. And now my problem is fixed: Picofly starts up when plugging in a charger or USB cable when the Switch is powered off. I'm a little concerned that I've being using my Switch without these capacitors for so long, but I haven't had issues in years of use, so I don't know if they're worth the risk of replacing.
My situation is pretty unique, but if someone's installing a Picofly by soldering directly to the board and runs into the same problem as me, double check that there are no bridges between any of the components around that 3.3V capacitor, either from solder or flux / IPA, because it appears that can prevent the Picofly from starting up when the Switch gets plugged in while powered off.