I don't understand what nginx has to do with this. nginx's language is just that: a language. If Lambda@Edge would support nginx language, you would still be invoking a lambda. But I'd must rather write some header rewriting code in, say, JavaScript than nginx's language.

Then why a (Turing-complete) programming language and not a config language? Simple: because a programming language gives you infinite more power than what any config system could ever provide. E.g. I could compare the origin IP, determine the country, and change the response based on that. And there are plenty of use cases like this which you cannot express in a config language.