My question is why is it so slow / is POST an async process on a "out of the box" express server?
You may be surprised to find out that this is probably first a browser issue, not a node.js issue.
A browser will have a max number of simultaneous requests it will allow your Javascript ajax to make to same host which will vary slightly from one browser to the next, but is around 6. So, if you're making 1000 requests, then only around 6 are being sent at at time. The rest go in a queue in the browser waiting for prior requests to finish. So, your node server likely isn't getting 1000 simultaneous requests. You should be able to confirm this by logging incoming requests in your node.js app. You will probably see a long delay before it receives the 1000th request (because it's queued by the browser).
Here's a run-down of how many simultanous requests to a given host each of the browser supported (as of a couple years ago): Max parallel http connections in a browser?.
My first recommendation would be to package up an array of requests to make from the client to the server (perhaps 50 at a time) and then send that in one request. That will give your node.js server plenty to chew on and won't run afoul of the browser's connection limit to the same host.
As for the node.js server, it depends a lot on what you're doing. If most of what you're doing in the node.js server is just networking and not a lot of processing that requires CPU cycles, then node.js is very efficient at handling lots and lots of simultaneous requests. If you start engaging a bunch of CPU (processing or preparing results), then you make benefit from either adding worker processes or using node.js clustering. In your case, you may want to use worker processes. You can examine your CPU load when your node.js server is processing a bunch of work and see if the one CPU that node.js is using is anywhere near 100% or not. If it isn't, then you don't need more node.js processes. If it is, then you do need to spread the work over more node.js processes to go faster.
In your specific case, it looks like you're really only doing networking to collect 302 redirect responses. Your single node.js process should be able to handle a lot of those requests very efficiently so probably the issue is just that your client is being throttled by the browser.
If you want to send a lot of requests to the server (so it can get to work on as many as feasible), but want to get results back immediately as they become available, that's a little more work.
One scheme that could work is to open a webSocket or socket.io connection. You can then send a giant array of URLs that you want the server to check for you in one message over the socket.io connection. Then, as the server gets a result, it can send back each individual result (tagged with the URL that it corresponds to). That way, you can somewhat get the best of both worlds with the server crunching on a long list of URLs, but able to send back individual responses as soon as it gets them.
Note, you will probably find that there is an upper limit to how many outbound http requests you may want to run at the same time from your node.js server too. While modern versions of node.js don't throttle you like the browser does, you probably also don't want your node.js server attempting to run 10,000 simultaneous requests because you may exhaust some sort of network resource pool. So, once you get past the client bottleneck, you will want to test your server at different levels of simultaneous requests open to see where it performs best. This is both to optimize its performance, but also to protect your server against attempting to overextend its use of networking or memory resources and get into error conditions.