There's a difference between the decision, and the way it was implemented.
I think it's pretty clear now that Pottechino offers a step up. So the decision to bring him in was the right one. Some of the critics seem to think you can never fire a manager unless he's performed terribly. There's no logic to that. If the owners are ambitious, you replace a manager if and only if you believe you can install a better one.
So I have no problem with the fact that the decision was taken weeks earlier, that it happened right after a great performance, etc. Al that was actually really smart. What wasn't smart is that it would have cost Cortese nothing to honour Adkins with a much warmer statement, program guide recognition, and perhaps even a farewell ceremony on the pitch. Instead of "whilst we acknowledge Adkins contribution..." he could have said: "Adkins was a great manager. A truly great manager. He brought us up 2 divisions. The club will always honour his performance and his time here. We'll give him a great send-off. But I believe I've found a manager who's even better and can take Saints to the very top of Europe.... "
That would have made the whole transition so much easier. He didn't do that. It's a real shame. But it was still a brilliant, courageous decision... ...and yes, it's time to move on.