For Fargate, the ideal scenario in my workloads is async background processing. Add tasks to SQS, Fargate pulls tasks off and does the job. Elastically scales up or down and lots of flexibility on machine specs. Ok with some failure rate.
For AWS Lambda, recently I like the combo of adding a Cloudflare worker in front and using it as an API gateway. More flexibility on routing, faster performance, and reverse proxy for free which can be good for SEO. And you get all the goodies of Cloudflare like DDOS protection and CDN.