As long as you didn’t ever write in https://browser.blockstack.com and instead used http://localhost:8888 everywhere, it will all be on your local machine. Using browser.blockstack.com you also store things locally in your browser as I understand, but you are relying on some server-side helper code run by blockstack. So your keychain ‘lives’ in your browser anyhow, but it will be indexed to the origin, so that could be why you needed to write your backup in again if you suddenly went to browser.blockstack.com instead of localhost:8888.
You do not need any Bitcoin in your account by the way. You also do not need a name to register. You do, however, need to verify at least one account.
Your problem is not one I encountered. I would check the DevTools console, and check for any errors there. I would also try to do a hard refresh first, as the developers have instructed other people. That is often
As another potential tryout: if you now actually did seem to use browser.blockstack.com anyway, and just want to get your registration in. I would try to open a Private Window, and with your keychain phrase, retry in this fully new environment.