IMO much will depend on both the initial priorities of the club, and then the roles within the new structure.
So far as the priority is concerned the club may feel it needs to re-establish some credibility in terms of the quality of football, their top4/CL and their value as a product for their commercial partners. If so then I do not think that a project is the initial means of doing so, and as such and potentially without the draw of CL football, the club will need to hire a manager who is able to provide a genuine sense that the club is serious in it's intentions to reform and go forward. This would be accompanied by changes to the match day squad in that it would no longer be used to provide development opportunities unless on merit, and a more mature / developed 1st XI would be put together in order to regain top4/CL within the current contract for the likes of PEA, Ozil, Mkhi & Lacazette. That does not mean there is no place for our development youth, but the priority shifts in the short term.
So far as the roles are concerned if the new manager is essentially a head coach, then the questions regarding transfers and club direction are handled by the CEO, Chairman, DoF and Head of Recruiting, whilst the manager concentrates upon the team training, match tactics etc. As such IMO the questions regarding relative experience in club management become less important if the club structure is strong and qualified in regards to relevant subject matter experts.
Under these criteria the likes of Loew remain in contention based upon his football credentials and achievements on the pitch, as do the likes of Enrique, Benitez, Allegri. That said Loew's involvement with the WC will pose issues - particularly if our season starts mid-July with the Europa quals.
Alternatively if regaining top4/CL is not the main priority (as opposed to the football and the ability to develop and utilise our youth), then the likes of Jardim becomes more attractive.
I do not count Ancelotti as he is seen as more of a fine tuner type rather than a (re)builder. Likewise I do not think BR has the credibility in Europe let alone the response from tour fan base. Arteta and TH have the draw with the supporters, but lack any experience as manager.
Tuchel and Nagelsmann have the reputation as the NextGen but have also yet to have any real success.
IMO the likes of PV ticks a lot of boxes both with the club and supporters, and the real question in regards to his suitability is more about the success and capability of the rest of the structure to support him. Likewise I think the likes of Fonseca or even Laudrup are also valid options, but I'm not sure that they'd be the best option under the current circumstances.
I suspect that Enrique may be the better option atm.
http://www.squawka.com/news/luis-enriques-football-philosophy-explained-from-dismantling-tiki-taka-to-msn-and-tactical-heresy/917022#zIqlWQLWJAeMz7pD.97
https://www.barcablaugranes.com/2016/12/22/14064182/tactics-key-difference-luis-enrique-barcelona-pep-guardiola
http://www.squawka.com/news/barcelonas-3-3-1-3-a-brutal-weapon-capable-of-overcoming-psg-and-storming-la-liga/912264#BSLLCemo0ZHdFqIg.97
https://fifaforums.easports.com/en/discussion/54570/aias87s-guide-to-a-forgotten-formation-3-3-1-3