ECC (and other methods) write the corrected value back to memory

That was my understanding (it corrects the error and writes the good value back to RAM), but now I'm not so sure! I imagine it must do that, otherwise a second bit flip would actually corrupt the RAM, and the RAM manufacturer would want to reduce that risk.

Regular ECC adds an extra parity bit for each byte. For each byte of memory, it can correct an error in one bit, and detect but not correct an error in two bits, so they wouldn't want a one bit error to linger for longer than it needs to.