My iPhone 5s is totally unresponsive, with a black screen.

Thanks a lot James, that does seem very likely. I wonder though whether new iPhone batteries (was not by apple tbf) come with some charge in them? Thus I should have had some response after replacing the battery before the charge depleted again?

They can come completely discharged or with a 30-40%. It depends from the seller. However you can check the battery with a common multimeter. But also if they come without charge once you connect it to the phone and then to the charger the phone should show the battery logo, charge it for few minutes and then turn on, if the battery it's not faulty of course. I think that your problem is what I told you because of the symptoms your phone shows. You can try another battery if you want but you will never find a real original one, because apple does not sell parts to anybody.