You are correct, the IP addresses will be replaced by the sending and receiving SIP proxy server IP address for that leg of session, although this isn't NAT as they are actually separate IP/UDP sessions.
SIP packets aren't treated any differently by routers, and if a packet needs to be routed between networks it will be forwarded with the source and destination IPs unchanged. SIP proxy servers are not IP routers though. SIP is just another application layer protocol transported over IP/UDP
When a SIP proxy server makes a routing decision it is actually looking at the destination SIP address (5) and not the IP destination address.
The client opens a UDP session to the first SIP proxy using its IP address as the source and SIP proxy server's IP address as the destination. The UDP payload is the SIP application layer traffic. The proxy server makes a SIP routing decision (different to IP routing) and opens a UDP session to the next SIP server and forwards the SIP traffic within that session. The IP and UDP layers for each leg are unique, but the SIP session end-to-end has the same SIP session ID and therefore is the same.
The proxy function becomes quite useful when making calls between organisations. Without the proxy server each organisation would have to have full routing information for each other's address space to make an end-to-end connection. There will likely be overlap so complicated NATs are required, which would also have to understand the SIP protocol to change any address at the application layer. With SIP proxy servers sat on the edge of each organisation the internal address space can be hidden. Clients only have to route to the SIP proxy server to reach any IP within that organisation.