The browser displays the URL from the origin that digitally signed the unmodified content.
A browser already doesn't show you what server delivered the content. That would be your wifi AP, cell phone tower, or ISP node. The internet has already long established that we can trust content without trusting intermediaries.
There are two elements that are important: integrity and privacy. The content integrity is protected via a digital signature, the "signed" part of "signed http exchanges". The signature proves that the document hasn't been tampered with.
Regarding privacy: The intermediary (a search engine in this case) already has the content being delivered as a result of crawling it. It also knows the user clicked on a link to get that content, and knows the user's ip address. Even without AMP or Signed Exchanges, the privacy situation is the same. Once the page is loaded, all further interactions with the origin are normal https traffic, so later requests are not different in privacy either.
What this enables, for search results, is the ability to load the bytes of the content before the user clicks a search result. If the browser prefetched those bytes with the origin's awareness, then the user's privacy with respect to the search query would be violated, making prefetch problematic. With this setup, documents can be prefetched while preserving user privacy and after the user clicks all browser behavior continues as normal from that point forward.