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"
By the time I joined, 18 months after development had started, a giant, complex, hideously tentacled software beast had been built that used every possible AWS service that the massive offshore team of developers could find to use.
It should have been built on a single Linux box by a single senior developer with Python and Postgres or nodejs or Ruby or whatever.
They went out of business after not too long and I couldn't help wondering if things might have been different if they hadn't spent a fortune building a giant money making machine for AWS, instead of making a web application on a Linux box.
Every AWS project I have worked on has had some significant work put into programming AWS instead of writing business functionality.