never buy a used phone that dosent turn on. especially iphones, because most likely even if you get it to power on the old owners apple id (icloud account) will still be logged in. and if thats the case the phone is a paperweight to anyone but the original owner. i would just open an ebay return and be done with it.
there are 3 different replacement screens qualities on the market. aftermarket lcd aftermarket oled and oem refurbished. in my opinion the am lcd is trash and dont waste your money. the am oled is the way to go for price vs quality. its hard to distinguish between am oled and oem oled. just do youself a favor and dont use the lcd.
if you replace the home button it wont work as a home button and touch id wont work. there is however a special charge port you can buy and install that will allow the home button function to work just not touch id. https://www.unionrepair.com/new-charging-dock-flex-cable-with-home-button-return-solution-for-iphone-7.html or you can bring it to apple for a screen repair and they will fix the home button in the process.
there are 2 cables coming from the screen that connect to the motherboard. one is for the screen one is for the home button. the smaller one is for the home button. look carefully and see if theres a small tear in the cable. if there is a new screen should solve your issue.