‘AI’ isn’t giving unsourced answers. Hastily cobbled together chatbots based on LLMs, inadequately prompted and without adequate research tools, are being put on the internet by humans and giving unsourced answers.
Well prompted LLMs with appropriate tools can do exactly what OP asked for. So AI can do this. That the AI-based products people are being given don’t is not a limitation of AI, it is a failure of humans.
Google overview frequently cites webpages that don't include the keywords of a query.
This is Google, the most used search engine in the world, made by a company that is at the forefront of AI development.
ChatGPT, Claude, etc,. constantly generate answers without sources and rarely cite anything verbatim. I'm not sure if they could cite something verbatim even if they wanted.
Essentially none of these tools are appropriate for information.