I'm in the UK and I got an invite early too. So they are not only inviting .US only people.
I would guess they invite those who use gmail who their bot sniffed out loads of domain related emails (not thay they would invade privacy like that).
Edit: I just used my invite now and it says "We’re sorry. Google Domains may not be available in your country. Sign up to get notified when it is available. Learn more Dismiss". The sign up page is
https://docs.google.com/forms/d/1viF3yadCXSyaw-Vya1nH3eoh26E_0lHs-UPTbu0UnU0/viewform
If I use my US vpn, It lets me login.
all 3 pages are equally clean, looks very simple and uncluttered, and sadly unlike the gmail invite only stage there is no limited free invites to others :/