Storing the history in memory also solves this half of this problem, and surviving a reboot is not necessary to ensure and audit privacy.
edit I'll stop. I can understand the flaws in this setup, I don't understand why you think it isn't possible/wouldn't solve the problem.
What actual evidence do you have that incognito mode is wiping the history at session end?
Implementing this is problematic because depending on the filesystem, it might not be possible to "wipe" a file by repeatedly writing over it. For instance, if it's a log structured filesystem. On the other hand, you might not even need to overwrite in some cases, eg. filesystems on SSDs with trim support. Also you don't need 35x overwrites. There's no evidence that even 1x overwrite can be recovered on contemporary hard drives.
No it isn't. You said nothing about a browser. You said storage medium.
an easy alternative is just to encrypt the drive using software-based encryption and just delete the key. Windows 10 has now stopped relying on hardware-based ssd encryption because it was found some of it was just garbage.
Edit: Additional caveats: some drives can move data in and out of the non-system viewable area and dead cells cannot be written if they're no longer viable so data can still remain there.
cat /dev/urandom > /path/to/mount/point/temp.bin && fysnc && rm /path/to/mount/point/temp.bin
is much more reliable and isn't dependent on your internet. dd if=/dev/urandom of=/dev/sdX bs=1MWiping the encryption key takes care of that problem, as well as being much faster.
Sometimes I do just that, I'll install Linux encrypted, and then reinstall later without migrating any bytes. My main concern is work related / personal finance documents being left over on a laptop. I've tested plenty of forensic utilities on my system after erasing files.
There comes a point where I simply won’t believe in magic. Whether it’s 3 writes or 7 writes, at some point there is not enough contextual information left on rigid magnetic media to recover any amount of useable information.