If Perplexity are visiting that page on your behalf to give you some information and aren't doing anything else with it, and just throw away that data afterwards, then you may have a point. As a site owner, I feel it's still my decision what I do and don't let you do, because you're visiting a page that I own and serve.
But if, as I suspect, Perplexity are visiting that page and then using information from that webpage in order to train their model then sorry mate, you're a crawler, you're just using a user as a proxy for your crawling activity.
If Perplexity are visiting that page on your behalf to give you some information and aren't doing anything else with it, and just throw away that data afterwards, then you may have a point. As a site owner, I feel it's still my decision what I do and don't let you do, because you're visiting a page that I own and serve.
But if, as I suspect, Perplexity are visiting that page and then using information from that webpage in order to train their model then sorry mate, you're a crawler, you're just using a user as a proxy for your crawling activity.