What difference are you alluding to?
Of course it doesn't matter, basically logging takes more resources than whatever routing is needed. (Especially because even if it requires running a Cloudflare worker it's just a NodeJS V8 isolate, very lightweight. As far as I know. But I have no idea of its costs.)
But AWS Lambdas are pretty pricey compared to a free nginx rewrite.
Kind of why CloudFront Functions we're invented for this use case, AWS Lambda@Edge is the more general purpose edge computing facility.
I'm not sure you're making a informed observation. "Spinning up a container image" is just fancy jargon/handwaving to refer to launching a process, which is something nginx already does under the hood in a myriad usecases without warranting complains from users.
> and queried for each request, don't you think?
I'm not sure if either you framed your sentence poorly or you don't have a clear idea about what you're talking about. In AWS lambda, you don't "spin up containers" at each request. You launch your lambda once it's deployed or when it scales up, and it stays up as long as you see traffic.
I’m pretty sure Lambda@Edge is not a container image like regular Lambda (why it only supports JS and not the broader set of Lambda runtimes) and is relatively local to the edge environments serving CloudFront anyway (why it is support in CloudFront the way regular Lambda isn't, and what the @Edge part refers to.)