CloudFront Distributions cannot pass the request to CloudFront Functions before sending to the origin. In other words, they cannot be used to modify origin request/responses. They can only modify the viewer request/responses. [0]
Only Lambda@Edge can help the scenario which I provided, which is also AWS's recommended solution. [1]
The origin request is your backend. The viewer request is the one you're requesting from the client. You want the viewer to have /api. You want the origin to not have /api.
Therefore stripping must be performed on the origin request, which is not supported by CloudFront Functions.
I have used lamdba@edge and our primary use case was to rewrite url (to add languages information for example) so I'm sure it works in viewer request events.
Personally, I did it with templates in api gateway, as at least then I'm not paying for the cpu time to do these sorts of manipulation. But yes, stuff like this made me really shake my head at 'the cloud'
