As the experience of working with cloud, after 1y 6m I can give a proper answer for my own question.
Yes.
There is a possibility to use API Gateway + Lambda for the entire app as the back end. But you have to manage your most of the app logic from the front end. On there you have to get a risk because the source code can be viewed by the public.
Keeping your all business logic in the client code is not a good practice. And keeping all the logic in the Lambda also not easy or cost effective. The reason is when you making a real world app, you will need thousands of functions. To do one task, you will have to call many functions (Then its a function run time). So it will be very expensive.
Best solution is hosting the backend on Elastic Beanstalk and front end on S3. If you have any heavy task ? then you can make Lambda functions for that.
Lambda is best for CPU bounded functions. But not to have all the application logic on it.