How can one reinitialize the HW.1?
I have a HW.1 too. I purchased mine, so it came with the recovery card. If you have your recovery card, you might be able to contact their support and ask how you could convert your security card's details into the 16 byte key

If you can't do this, then use the link you posted previously to generate a random key, and then generate your card. Remember to save both.

Delete the seed currently on the HW.1 by typing in the seed 3 times incorrectly.

Plug your ledger in and then try to update the firmware. Once it prompts you, put in your random key. If this doesn't work, try using their online site. It may not be as safe. IIRC you can download their special initializing software and mount it as a boot drive on a USB and do it air-gapped.

If it still doesn't work, I guess your best bet would be to contact their support.