Theoretically, you could also do the entire setup via Cloud Formation; but you can also do the same with EC2 instances and ECS.
What would I do instead? Set up a 1-n member autoscaling group, with rules to scale on load. Set up an ECS service which auto scales on load. Set up an ELB attached to the ECS service. Not as fancy as Lambda and AWS Gateway, but it will probably scale better, at a lower peak cost (at the expense of an up front cost of one server).
