No coach is satisfied with the legacy of the previous coach. Antonio Conte is a coach who has quite a good reputation and he will not immediately accept a contract if it is not followed by a large investment from the club. Maybe Conte will retain several players who he thinks are worth bringing together, apart from that Conte's philosophy will also make Napoli's game scheme change completely, he really likes using a three-defender formation which means he will focus on improving the back line first.
The legacy of a previous Coach shouldn't matter a lot to a new coach especially is such a coach failed to get the team to victory but there are lesson to learn from the failure because from there the new coach may get to know about things that doesn't work with the team and doing them aswell, but if the coach was a successful one, he's got a legacy to protect or carry on so it becomes important to the new coach at that point.
In your words, Conte's philosophy is going to be a game changer for Napoli and I believe this to be true going by Conte's legacy back at other teams where he has managed before he got into a contract with Napoli. Just like every other coach, your success has got a lot to do with the quality of players in your team after which your management skills or your ability to manage these players well enough becomes important too aswell, because it's possible you have got a good number of good players yet you are not able to manage them well enough to produce results.
Napoli have got a good number of players who can be considered to be relatively good with their performance of which I want to believe that with good management wherein they get a brilliant strategy and tactical support, they are most likely going to do better than they have done previously and it appears like Conte can get it done with the Italians in the coming season as their head coach.