So, I could login to my new SharePoint site with the Administrator account when I was logged into the server, but not when logged into another machine.
I was able to pull up Central Administration from the other machine.
After much puzzling, it looks like the SharePoint service account for my site isn’t correctly configured. When I switched the service account to the same one Central Administration is using, I could log in just fine.
Turns out the service account hadn’t had the SPNs set:
setspn -s HTTP/FIMServerName domain\SPService
setspn -s HTTP/FIMServerName.domain.com domain\SPService
setspn -s FIMService/FIMServerName domain\FIMService
setspn -s FIMService/FIMServerName.domain.com domain\FIMService