Conte would be good again, just based on what he managed at the Euro's with a depleted squad. My only thing is would he be willing to take over a rebuilding project, and if he would be willing to play the optimal formation to utilize the talent available. This is Italy's most talented group of natural wingers/wingbacks in a long time, and any formation that doesn't get the best out of them would be a real waste. There is no reason Italy couldn't make a deep run at the 2020 Euro's with the right coaching.
GK-DEF would be the least of my worries, there is a lot of depth, and quality available there. Verratti and Jorginho provide a strong base to build around in the MID as-well. If Pellegrini, Gagliardini, or Cristante can take that next step they would provide some much needed energy and grit alongside those two, there's also the likes of Barrella and Mandragora who should break through before the Euro's.
The biggest issue right now is striker.
The next coach really needs to integrate Balotelli back into the squad, there is no excuse at this point. He's been great for Nice, and has a good goal scoring record for Italy too.
Immobile can't replicate his Lazio form for Italy, I don't know if that's down to tactics, formations, or just mentality but its a glaring issue. Italy would have been up 3-0 on England if Immobile takes his chances in the first 25 mins, obviously I give him credit for getting into those positions but he's just been extremely wasteful. He missed numerous chances in the 2nd leg vs Sweden as well. It's just not acceptable at this level.
Needless to say, this is a huge decision and I'm praying they don't fuck it up.