This seems to help mostly against servers improperly using cookies, servers improperly logging request content, and users improperly uploading HAR files that include bearer token.
And anyone who does those things improperly will also implement WebSession improperly - like not bothering to keep track of nonces - so what does it really gain us?
edit: just broadly on "pwn your machine vs pwn your DNS" - overall, in the general case, machines are much much easier to pwn.