iPhone 8 Plus home button stopped working randomly

This problem usually occurs from a small tear in the home button flex from the screen. But you have tried another screen so have ruled out the screen flex. This leads us to the home button assembly itself. You will probably need to put the home button assemble under a microscope and see if you can find the fault, it may be a small tear in the flex, damaged trace on the connector or the turtle IC Either way they are not easy to repair.

It seems strange that the problem did not occur straight away after the screen change. I would also make sure it is getting power to the connector.