Well obviously you don't need to maintain a background thread and a connection so you would be able to send out much more data using the same resources by offloading the TCP workload to the proxy.
If the service you are sending data to is unavailable, you wouldn't get any backpressure from it.
Think of a monitoring or log system where you might want to send out millions of datapoints but you can afford to lose some, in return you don't have to worry about the system not being reachable.
Until you get some decent amount of load on the system. Then the background thread will still make progress. And the UDP/Proxy approach will only have failing requests, because the loss of any single UDP packet will make the request fail (there are no retries).
I can only see being interesting if you have a workload which requires sending small requests that fit into one or a couple of datagrams.
If the service you are sending data to is unavailable, you wouldn't get any backpressure from it.
Think of a monitoring or log system where you might want to send out millions of datapoints but you can afford to lose some, in return you don't have to worry about the system not being reachable.