If there’s some sort of natural limit to how many simultaneous connections they can handle, why can’t they just return some 4xx error code for connections beyond that? (And have clients implement an exponential back off?)
Or if that’s too difficult, the load balancer could keep track of some maximum number of connections (or even requests per second) each backend is capable of, and throttle (again with some 4xx error code) when the limit has been reached by all backends? This is pretty basic functionality for load balancers.
You’re going to need actual congestion control anyway, when the number of client connections is unbounded like this. Even when your servers aren’t crashing, what if the client apps whose clicks you’re tracking becomes suddenly more popular and you can’t handle the load even with all of your servers up?