"without blowing $250,000/month on their AWS bill". The point is that you don't need AWS for this! You can use Docker to configure much, much cheaper/faster physical servers from Hetzner or similar with super-simple automated failover, and you absolutely don't need an expensive dedicated OPS team for that for this kind of simple deployments, as I read so often here on HN.

You might get surprised as how far you can go with the KISS approach with modern hardware and open source tools.