I've implemented quick exit/escape buttons for a few organisations over the years and have spent a bit of time thinking about the limitations.

This pattern is definitely better than most and it is refreshing seeing they put some resources into it. In my professional experience, organisations often chose the "a link to another site like google is fine" option to save money and time while still getting to boast about their security culture.

One thing I have not found much research on however, but would love to hear about, is the effect of these kinds of patterns on the user's speed and choice of actions and how that effects outcomes. What I mean by that is, say someone is visiting the site on their phone and an adversary walks into the room. Most people these days know the fastest way to leave a page at short notice - maybe the home button/gesture, maybe swipe to another open app. Does having a big red button that introduces a new choice help them, or add to the cognitive bandwidth needed to handle the situation?

Remember, by definition the type of situations that this component is intended to help with are going to be stressful and likely have little to no warning; the person is going to walk in the room and the user has moments to act.

What is going to lead to measurably better outcomes; a big red button that the user needs to read, understand and move their finger/hand to, or their own knowledge of their own device's most efficient escape mechanisms?

This isn't meant as a criticism of the component. I am just genuinely curious as to what the best tool to assist folks in this situation is? We are talking about real people with real fears and the possibility of very bad outcomes.