A mid sized web site with 5000RPM would be 219 million of requests per month: so 766 USD/month just for API GW, without data transfer, lambda, static/cdn or DB costs.
If you expect a steady load of hundreds of requests per second - maybe API Gateway + Lambda is not the best solution. The use-case in the article did not have such load targets. Lambda is not a silver bullet..
In this scenario, the savings come from time + resources spent on configuration management, system admin and troubleshooting, and server security.
Also if you have servers you are probably auto-scaling redundant instances in multiple availability zones and then getting that high CPU utilization is unlikely.