When user enters a prompt, the backend may retrieve a handful a pages to serve that prompt. It won't retrieve all the pages of a site. Hardly different from a user using a search engine and clicking 5 topmost links into tabs. If that is not a DoS attack, then an agent doing the same isn't a DDoS attack.
Constructing the training material in the first place is a different matter, but if you're asking about fresh events or new APIs, the training data just doesn't cut it. The training, and subsequenctly the material retrieval, has been done a long time ago.
The problem isn't the tool, it's the user: they don't know if they're getting good code or not, therefore they cannot make the prompt to improve it.
In my view the problems occur when using AI to do something you don't already know how to do.