I'd say it is convenience, but I don't know the lamda_http API. Axum has a lot of nice abstractions and a big ecosystem that makes building HTTP servers very easy. On the other hand, using the Lambda API directly instead of a web framework is considered good practice for bigger serverless apps.
I'd say that depends on your concrete use-case and how complex you expect your backend will become. I think the decision between serverless or traditional hosting is mostly one of convenience vs. flexibility (besides vendor-lock and pricing, but these may be out of your hand). If you are building a simple RESTful API service on top of DynamoDB or RDS, don't see why you shouldn't use serverless (if you are dedicated to giving a lot of money to AWS, that is 
). But with everything that is as highly abstracted as Lambda's computing model is, you may find yourself confronted with a feature request that doesn't fit that well into the serverless model.
By the way, I think there's a middle ground between EC2 and Lambda you might wish to consider, EKS. Kubernetes gives you ridiculous configuration and scaling capabilities while still being as flexible as to allow you to self-host everything you might need for a modern web service in your cluster.