No, logon server will be picked at random from a domain controller in the same site as the client computer (providing the subnet the client computer is in is asociated with the site in AD Sites and Services). I’m more concerned about your general AD health. What you described with the DC is obviously not normal. I’m going to say check DNS because so many times it’s DNS

For your domain controllers I would have DNS on each DC set this way.

DC1 in site 1
Primary DNS: DC2 in site 1
Secondary DNS: DC1 in site 2
Tertiary DNS: 127.0.0.1

And no public DNS listed on the DC NIC or client NIC.

Follow the same pattern where you point to the other DC in the site first, a DC in a different site second, and 127.0.0.1 third. You can have more than 2 DNS servers if you go to the advanced button in TCP/IP settings and then the DNS tab. You may know this already but we get all levels of skill in the forums, and others may not.

I would bring your other DC back online and see if we can troubleshoot that.