What you describe is basically how persistent http works, is it not? Even a persistent connection terminates at
some point. Which web server does not work like that?
I guess you could send the connection header on draining, but anything less than what the big servers do is bound to cause some compatibility problem with some niche client somewhere. I can certainly see why a web server with millions of installs would be reluctant to change bevaviour, even if it is within spec.
I can only guess at the use case here, but maybe something designed from the start as a stateless proxy and not a general purpose web server would be a better fit.