----- SOLUTION FOUND -----
Posting a solution here for any CTRL+F heroes

Retroarch Quick Menu > Core Options > HLE BIOS = ON

I don't know what the setting even does, I'm just glad autoload actually works now.
Gonna put some tags below for search engines to pick up on:

lr-flycast, flycast, autoload, auto-load, auto load, crash, savestate, black screen, saves, saving, loading, loads

Normally you'd compile it manually and add it as a separate emulator, but the current version doesn't compile on current RetroPie (Pi3/Pi4). You may wait until the new version of RetroPie (based on the Raspi OS bullseye) is released and the standalone flycast will be installable.
I've started working on a RetroPie module to integrate it, but due to some rendering bugs and the aforementioned incompatibility, I didn't add it to RetroPie proper yet.