Liverpool changed their structure to suit Klopp.A good structure and being well run is important but it is worth pointing out that Liverpool weren't a particularly well run when Klopp took over, the year before they had sold Suarez and replaced him with Balotelli and Lambert and the summer before sold the best English talent of his generation to a rival. What they did was get themselves a brilliant manager which gave them the room and time to sort themselves out.
The issue for United is that badly run clubs make bad managerial appointments and we have done that every time (it's nonsense to suggest Mourinho didn't fail despite him aiming for short-term results and still never making us look like a coherent team despite scraping a few wins against shite teams in the Europa League). If we do somehow make the right appointment next time, personally I would go for one of Rose or Nagelsmann for their potential and ability to bring more modern ideas, then they would solve a lot of problems and would potentially push us to sort out the structural problems.
Nagelsmann or Rose would force a change in structure but it remains to be seen if they can replicate Klopp's success. But i'd be much more confident if either/or, were in charge due to their potential and body of work which hasn't gone unnoticed.