Author here. Great points, I agree 2 extra round trips are not strictly necessary.
The advantage of P2C is the simplicity that comes from its stateless/just-in-time nature. If you cache load, either by servers broadcasting or including a load indicator in the response, you have to consider cache invalidation and TTLs. The longer the TTL, the higher the odds of a thundering herd (all clients think one server is under loaded and proceed to overload it). If you're undertaking this complexity, it may be better to go with a proxy, which can have other benefits too (connection pooling, no client cooperation required). Like all such decisions, it depends on the situation.
The advantage of P2C is the simplicity that comes from its stateless/just-in-time nature. If you cache load, either by servers broadcasting or including a load indicator in the response, you have to consider cache invalidation and TTLs. The longer the TTL, the higher the odds of a thundering herd (all clients think one server is under loaded and proceed to overload it). If you're undertaking this complexity, it may be better to go with a proxy, which can have other benefits too (connection pooling, no client cooperation required). Like all such decisions, it depends on the situation.