> I've seen just API Gateway costing more than entire infrastructure costs of similarly sized websites.

Do you mean Api Gateway + the lambdas?

API Gateway is $3.50 per 1 million API calls received plus data transfer ($0.09 per GB or $90 per Terrabyte). What kind of API were they serving that was racking up those large bills?

Typically you would serve static content off s3 + cloudfront, and just the API off the API gateway/lambdas.