Two entirely different issues.

1. Giving people warning.
Yes, they will look for a browser cookie to see if you have a login session (which is associated with your account), and will pop the warning about the sign up bonus based on that. I'm quite sure that is simply an attempt to proactively set expectations with the rabid circles-n-arrows mobs who woulds otherwise cry foul and wave screenshots. Giving a warning probably will just reduce a little of the noise their CS agents have to deal with from aggrieved applicants who get the card but no bonus. It's entirely appropriate, and pretty easy to do. Good on them, it shows they're willing to make some effort to guide noobs (and to take the edge off the CS complaints).

2. Actually getting the sign up bonus.
This is certainly not going to be determined - at all - based on what cookies your browser returns to their website. Amex is not stupid. They're going to make their assessment based on your credit history with them, which they have easy access to and can't be easily tweaked by anyone who knows how to launch a web browser. You can delete cookies, switch browsers, and play all the browser games you want, and that's not going to make any difference whether you get the bonus or not.