I'm sorry about your bones, but you can already lie and misquote on the Internet!
Although I suppose the frame around the quotes that this service provides does give the text an air of authority.
In the happy path (where the quoteback is genuine) the browser could indicate to the user that the quoteback is genuine and the link would take the user to the normal website if it still has the quote, otherwise to the archive snapshot.
In the unhappy path (where the quoteback is disingenuous) the browser could notify the user that the quoteback is disingenuous.
If the snippet does indeed come from the website in question, the website will return a signature which the quoter can embed along with the snippet on their own website.
The signature would prove that the content came from the same person who controls domain X (as attested by the CA fo your choice). The user agent can display all this information where the content is quoted.
Sounds nice, but my gut feeling is that you wildly underestimate how far people will go to work around anything that could change their beliefs.
This seems to hold true on all sides of the political spectrum, in art as as well as in science and the only difference is what beliefs people stick to.
As a deeply religious person this might come of as really ironic and the irony is not lost on me: quite the contrary and for that reason I've thought about it multiple times.
The man agrees, "Yeah, corpses don't bleed."
The doctor pokes him in the arm with a needle and blood wells up.
The man looks at his bleeding arm in astonishment and says, "I'll be damned! Corpses do bleed!"
(This is a true story. I think it's in "The Man Who Mistook His Wife for a Hat" IIRC.)
Alternative: Isn't there some way to save an HTTPS request in a way that others can verify that the webpage was signed with an authorized certificate?
Future: I wish authors would sign their content directly rather than depend on TLS and certificate authorities. P2P networks do this well (because they have to) but it hasn't caught on in the rest of the web.
That sounds a lot like "Signed HTTP Exchanges"[0], which has some support in Chrome but doesn't automatically provide a way of checking quotes in web pages.
[0] https://wicg.github.io/webpackage/draft-yasskin-http-origin-...