If cargo run downloaded a new toolchain, it means there wasn't a nightly toolchain installed already[1]. At least in my test, rustup toolchain list did indeed show the override after the install occured
The problem is that the nightly toolchain for aarch64 is installed but the override thinks this is an x86_64 host, which is not as this is an Apple Silicon box.
And obviously line 1 is wrong. I believe that's because I migrated from an Intel-based MacBookPro to the M2 Max one and that line must have been there since forever.
I just run the following to set it up correctly: rustup set default-host aarch64-apple-darwin
And now it is working as expected Thanks a lot to all of you!