Round numbers are tied to a date time in a loose sense that each participant should be emitted a share for every time period since the genesis. A single node emitting its shares early or late wouldn't affect the network - only a threshold number of nodes doing so

I believe this depends on some blockchain, and you can't move forward without a consensus from a majority of nodes.

It's not consensus per se - each node releases a share of a secret and if any node gains a threshold number of shares, they can use interpolation to generate the signature and thus the randomness. Different nodes could even collect a different set of shares to reach the threshold and they'd still generate the same signature

To clarify: there's no blockchain involved.