Display replacement. Will same home-button work on new display?

Home button and taptic engine should be fine, taptic as you see is under the battery so it's not part of the screen.

For the home button as long as you are careful in removing the home button there should be no problems.

I recommend refering to this guide for the home button part so you know how it comes out:

Sostituzione schermo iPhone 8 Plus

I have had some cases where a replacement screen had a defective home button extension cable so look out for that if you have issues with the home button not working properly.