Exactly, swapping the original button to a new front display assembly should work perfectly fine. Just take care removing the original home button, low heat is your friend. And if you're using your original lcd shield take care removing that as well. Again low heat and fine tip tweezers under the little metal piece adhered to the socket.
You should actually be commenting in a home button step (such as 30), so you get the right visibility. But if by front plate you mean the display (digitizer, lcd, and front glass) then no, this will NOT fix touch id problems. If the problem is damage to the home button out it's attached cable you are out of luck, as those are unique to each iPhone. However if the ribbon cable (the one attached to the large shield) is damaged, then that is replaceable. A new lcd shield should come with the home button ribbon attached, or you can buy the ribbon separately and just replace that. The shield is much easier to replace and about the same cost.
When replacing, make sure the proximity sensor is replaced correctly and seated fully inside the plastic bracket. Not doing so can cause loss of functionality in the sensor preventing the lcd from turning off when it's supposed to during calls etc.