Regarding the OP
1-I think primary approach will be sounding a former manager to give him full authority. I think if there is any time Pep might consider us it will be now. He is losing his touch with City, already overstayed there, there might be a ban and contract will end next year. Unless he will go to Italy there is nothing else to do for a coach of his type. He doesn't suit NT football. Also sounding Lucho if he isn't interested to continue with Spain. Both are strong character coach who are club legends. Both have done a major retooling job. Both will get the respect and keys of the team. Laporta might have a chance in convincing them. But he won't do that if they are interested in continuing where they are.
Is it going to happen? will give it a 10% chance that one of them will say yes. But we need to try.
2-Second will be a big name coach, and there is no abundance of those. I will say Zidane,Simeone & Klopp are currently the only big name coaches out there , excluding aforementioned Pep . 2 of them coaching rivals and will never come. Would Klopp even consider us (even under new board) ? I think it is same answer as above, only if he feel his job is done with liverpool.
3-The next thing close to a big name is Allegri, who is a EV type of coach in terms of style, although on a higher level. He has helped to bring Milan back, helped to take Juve to the next step. But he isn't the one to deliver anything close to Barca DNA or a CL aspiration. Pochettino is another name, who always talks about us with true hatred. But I think he will come if we called him. Can't see it happening with any of those names.
4-Club legend, and the problem is that we are running out of those. Koeman is the most proven name, Xavi is the upcoming one. Gio is probably the only one that might make sense tbh.
5-Then there is the high risk high rewards: New young coaches like Gallardo, ten Hag (not really young, but new) if he speaks some Spanish words. Problem with this approach is demanding respect from locker room.
6-There is some other bad choices to mention: The old names, the Blanc,Mourinho,Ancelotti, Wenger of the world. The 50+ Liga coaches who wants a chance in here. Hopefully not, but with this board they might do it.