iPhone 6s no display not entering dfu mode but charging the battery

first get ahold of a copy of 3uTools, it’s a great info and diagnostics tool and it’s free.

it’s also a good way to see if the phone is really booting up or not - with or without a screen.

as you say you’ve tried a good screen and it still doesn’t appear to boot. have you tried the bad screen with a known good phone to see what level of functionality it still has? have you recently confirmed the new screen is indeed a good screen?

in your shoes, i’d definitely try more than one screen on the phone in question.
i would try seeing if i could get it to boot into DFU mode, maybe flash to the latest ISO (all depends on if you need to preserve the data or not)
i’ve had phones with motherboards that were fully functional but would not display video. i’d guess damage (physical or electrical) to the video circuitry on the mobo (i can’t diagnose to that level, so i stop with that conclusion) and unfortunately i declare the motherboard unusable. supposedly there are services that can fix this sort of thing, but i haven’t chosen to engage any. depending on the nature of the damage this might be a possibility