The design is very good. One suggestion: Consider having the LLM use bullets for the longer summaries, such as "KGM (Kaley) — The Plaintiff Takes the Stand." This will relieve the reader of facing a wall of text.
Also, regarding "Every claim sourced from live web search": For me, that phrase created the expectation that every claim made by the various participants would be itemized. Instead, the system seems to be capturing the claims in news reports of testimony. That's not a bad thing. Most people probably would prefer the focus on key claims reported in the news, as opposed to a long list of all claims, many mundane. But it's worth thinking about whether a different tagline phrase would better communicate what's being delivered.
@laurex are you the creator? Can you tell us some more about how it was built?