Update: after many attempts I have gotten it to work by doing the following:
Disable system integrity protection (boot into recovery > terminal > csrutil disable)
Run sudo rm -rf /System/Library/AssetsV2/com_apple_MobileAsset_MetalToolchain/
Enable system integrity protection (boot into recovery > terminal > csrutil enable)
Install the toolchain
Manually mount the file at /System/Library/AssetsV2/com_apple_MobileAsset_MetalToolchain/*.asset/AssetData/Restore/*.dmg
As soon as I mount it, it works. Unmounting it makes it not work again. If any engineers would want to look into why that's happening, I'd be happy to help them debug this.