First: this isn't true. There are plenty of Ashokan inscriptions from ~150 years after the Buddha's death, and they speak of an already-existing Buddhist orthodoxy.
Second: there's plenty of archeological evidence of both the Buddha and Muhammad; the lack of written documents has more to do with the cultures they were in at the time. You don't need to drag other religions down to engage with the evidence for the historical Jesus.
Yeah, if you assert a document is not a document, sure, you can say whatever you'd like.
The news here is that AI has too much trust in the internet. The first time I allowed tool-calling, it started googling up some nonsense instead of thinking... But I think at least it's possible for the AI to evaluate the quality of the source - you just have to ask for an analysis, and you'll get a reasonable evaluation. With humans, something like that just doesn't work - they'll get aggressive or might even start throwing bananas...