I'll poke the bear:

As a user, the browser is my agent. If I'm directing an LLM to do something on a page in my browser, it's not that much different than me clicking a button manually, or someone using a screen reader to read the text on a page. The browser is my user agent and the specific tools I choose to use in my browser shouldn't be forbidden by a webpage. (that's why to this day all browsers still claim to be Mozilla...)

(This is very different than mass scraping web pages for training purposes. Those should absolutely respect robots.txt. There's a big difference between a user operated agentic-browser interacting with a web page and mass link crawling.)