No, the tracking across sites will stop. None of those "silly" laws prohibit the use of e.g. session cookies.
edit: looking at the code I even see references to a “Reject All” button as well as an XHR request to a geolocation endpoint[1].
[0]: https://dl.dropboxusercontent.com/s/vw2khgjnyrnqucb/f
[1]: https://geolocation.onetrust.com/cookieconsentpub/v1/geo/loc...
And that just brings up the question again: why show the popup to non-EU viewers if the only option is to accept the cookies. It isn't even effective as a form of protest because those viewers have even less influence over EU regulations than EU citizens.
No, they care about being minimally compliant and laws vary.
A number of places have legislation that just says "users much opt in" but the EU has legislation that says "users must opt-in, but cannot be forced to opt-in by there being no opt-out option".
They don't care about your privacy at all, no matter where you are, if any claim otherwise is made it is a lie.
A bit of bad design there, not failing safe, if there intention is to be compliant with stricter laws, because a failure in an external dependency makes the site less compliant.
If anything, putting such wildly non-compliant popups should make you more liable to punishment, because it undermines an argument of ignorance.