For this first implementation, I don't see anything ever added to the queue. Am I missing something? New task is added to the queue if the queue is not empty only, but when the queue is empty the task is executed and the queue remains empty so in the end the queue is always empty?
If there is some kind of cooperative multitasking going on, then it should be noted in the pseudo code with eg. async/await or equivalent keywords. As the code is, send() never gives back control to the calling code, until it completely finishes.
let send = (payload, callback) => fetch(...).then(callback)
fetch() returns a promise synchronously, but it's not awaited.
[1] i.e. backpressure, which would actually be the ideal way for the server to implement whatever rate limiting it wants, but we are assuming here that the server has a less than ideal interface.