I just meant for servers to maintain a local chain. Cross-server checking would be free, sort of, as clients that talk to multiple servers would inject the chain state from the first server into the second via their nonces.
At scale, a time server's DNS address will likely either be a reverse proxy or a multi-valued A record. The coordination amongst those separate physical servers that are serving requests for the same DNS name would become a scale challenge if they needed to share state.
But they wouldn't need to share state. They could run independently, and any client that wanted to tie their clocks together could request the signed time from one and then send it to the other, causing it to get logged in the other server's chain.
