simple_chat: Agent appends onto text from a prior turn instead of adding new text at end of chat when responding to user

_↴ Ported from [flutter/genui#982](https://github.com/flutter/genui/issues/982) — originally opened by [andrewkolos](https://github.com/andrewkolos) on 2026-06-24._
_Original labels: front-line-handled_

---

Reproduced on ffde58489f8e976288feb2ef4d6110515c58d269. Possibly related: https://github.com/flutter/genui/issues/517

Notice the part of the initial text starting with "Acknowledged!".

**Expected behavior**
A clear and concise description of what you expected to happen.

**Additional context**
Add any other context about the problem here.