The first concern is the most legitimate one: can I stop an LLM from training itself on my data? This should be possible and Perplexity should absolutely make it easy to block them from training.
The second concern, though, is can Perplexity do a live web query to my website and present data from my website in a format that the user asks for? Arguing that we should ban this moves into very dangerous territory.
Everything from ad blockers to reader mode to screen readers do exactly the same thing that Perplexity is doing here, with the only difference being that they tend to be exclusively local. The very nature of a "user agent" is to be an automated tool that manipulates content hosted on the internet according to the specifications given to the tool by the user. I have a hard time seeing an argument against Perplexity using this data in this way that wouldn't apply equally to countless tools that we already all use and which companies try with varying degrees of success to block.
I don't want to live in a world where website owners can use DRM to force me to display their website in exactly the way that their designers envisioned it. I want to be able to write scripts to manipulate the page and present it in a way that's useful for me. I don't currently use llms this way, but I'm uncomfortable with arguing that it's unethical for them to do that so long as they're citing the source.