Refactor HttpClient stream setup

closes #16510 
If the bytes contain multiple responses the parse only parses the first one and keeps the remaining bytes in its internal continuation state.
I see now that my struggling to work around the limitations of the FlexiMerge (#16565) has actually messed up my initially correct design of properly draining the parser by switching back to the methodBypass and staying there until the parser state has been properly drained.
I'll get that fixed right away.

> What if the last one is incomplete?

Then the parser will produce a respective error output in `onUpstreamFinish` called from the completionHandling.
However, I need to get that fixed as well because emitting from the completion handler is not reliably possible (#16565).