I wouldn't want the job of any of the MDs that we're talking about - MH, AM, and JF - and I really doubt I could do the job a fifth as well as they do. But, yes, their main job is managing and directing (goes with the title!) and that's where they are required to excel. Whether and how their way of working and character filters across the teams they run and though to interaction with the customers who interact with that team isn't necessarily the most-sought, most-required attribute of the job, even though it's the attribute which we as customers see.
The "main job" of an MD is to provide Leadership, as distinct from management. He is also accountable for the company's performance - judging by the current level at
GWR▸ , suggesting he does it "well" is a highly subjective judgement.
My MD isn't a huge fan of dealing with the public, but he respects them as his customers so he makes the effort to engage with them from time to time - it goes with the pay grade, and they respect him in turn because he makes the effort. It also gives him the sort of insight into perception of his Company that a thousand Board reports never could. Why join a public facing organisation if you never want to face the public?
An MD will generally set the tone for organisational culture as well and this permeates throughout the company -perhaps it's legitimate to suggest that, given his reluctance to leave his office and meet his customers, there might be a connection between that and a "filtering across his teams" with the outcome being the appallingly low levels of customer service which his Company provides......?