Chosen Solution
I have just replaced the dock connector (and this corrected the problem of impossible shutdown of the phone) but now the home button is no longer functional. I am pretty sure it worked before. Any clue or suggested investigation before I commit to the difficulties of replacing the home button and/or home button flex circuit ?
Check the small flex cable connector of the home button, that is connected to the dock connector flex cable you replaced. Make sure it is secured in the connector.