sshd -ddd complained bad owner or modes for /mnt/c/Users/user initially, so I did chmod 755 /mnt/c/Users/user. Again, sshd compalined about the permission stuff for /mnt/c.

I turned off strict modes to get a simple workaround. Though it's not a good way to do it, this and the metadata thing together can get public key auth work on the current wsl.

Maybe keeping two copies of .ssh is in fact the best painless solution. This is what I feel after a whole struggling.