No automatically, but it can be done manually. And it should be done before you put this new primary node on-line.
The case above is a special case, where you lost ALL your primary nodes.
If they are recoverable, then you should make them on-line again and only then add the new primary node. In this case it can copy the database automatically.
We are talking about disaster recovery. We can avoid these by having something like 3 primary nodes and having at least one of them on a cloud provider. On this situation it is highly improbable to loose all 3 at the same time.
There are basically 2 scenarios:
Scenario A: If you need the nodes to synchronize locally even when the internet is down:
Use 2 primary nodes on your local network (computers or raspberry pi / orange pi / nano pi...) and 1 primary node on a cloud provider. Make the secondary nodes to connect to the local nodes.
Scenario B: The secondary nodes are on the field and require internet connection to synchronize:
Use 3 primary nodes all on the cloud, but use one on each location, and preferably one on each provider (Linode, Scaleway, Alibaba, to list the less known...)