The lie is in their documentation - they claim to use the PerplexityBot string in their user-agent: https://docs.perplexity.ai/docs/perplexitybot.
I think it is OK to use a different user agent for page retrievals made on demand that a user specifically requested (not to include in the index, just to answer a question).
But... I think that user agent should be documented and should not just be a browser default.
OpenAI do this for their crawlers: they have GPTBot for their crawler and ChatGPT-User for the requests made by their ChatGPT browser mode.
At the very least, I do think that having an entire page in your docs about the user-agent strings you use without mentioning that, sometimes, you don't use those user agents at all is fairly misleading.
In the blog post, this is not what is happening. It is merely feeding the webpage as context to the AI during inference.
You are all confused here.