Hi DinosaurShadow,
That may not be entirely correct. I think you still need to link your Xbox Live account to your EA account even if it is not an Xbox Live Gold account. When I was creating an EA account for NFS Payback I did need to be online and signed into my Xbox Live account. Creating the account through my Xbox One automatically linked it. I can't remember if I also had to do an e-mail confirmation. I think I did, but once that is done you should be all set.
The reason you need to be logged into your Xbox Live account and have it linked to your EA account is that the game requires a constant online connection. (That is, if you are talking about NFS 2015.) but you shouldn't need Gold.
Also, NFS Payback is playable offline without an account, but it seems that now that I have linked my Xbox Live account to my EA account that my save file now requires me to log into my Xbox Live account. However, if I start a new game offline and not signed into an account, then I do not need to log in.
If you still have difficulty I would recommend attempting to log into the game using your Xbox Live account and if it asks you to create an EA account, see if there is an option in the game when you start it up to log into your EA account. If there isn't, but there is a screen to create a new EA account, I would try that.