To be fair, if they had a AWS Solution Architect involved they heavily push you down this road and if they manage to get in management's ear they'll push the idea that server-less AWS features is vastly cheaper.
If you're only responding to a handful of requests that's true, but once things ramp up you get "nickel and dimed" for everything: API Gateway requests, lambda execution time, DynamoDB read/write units, CloudWatch logs, outgoing data, step function transitions, S3 requests.
I understand all those services cost money and they shouldn't be free, but I question if paying all those micro-transactions is worse then paying for your own VMs, especially once your customers complain about the cold starts and you think you can fix it with "lambda warming"