The role of a Commander in COIN or COIN-like operations is not identical to his role in MCO; he effectively has less to do and thus more time to devote to minutia if he's inclined to do that. This can have ramifications down to the lowest level, not just to the Staff. Due to the operational tempo, the Staff also has more time...

Then there's the trickle down question; the effect on the working troops. In the first cited unit below, Platoons and Companies just worried about the job at hand -- Brigade was 'the rear.' In the second, they did the job but always with an eye toward appearances at the neglect of performance as Brigade was always in their thoughts -- thus teaching a lot of Lieutenants and NCOs some really bad habits...

I served under two US Brigadier Generals that mirrored your examples. One operated as did your British commander, to the proverbial 'T.' His standing order at night was "Wake me only if all three Battalions are in heavy contact." People were expected to inform him if changes occurred between the Tuesday and Friday A.M. scheduled briefings.

The other was much more into your American model, to the extent that he was insistent that not only did he want company efforts run by him before execution, he wanted reports to the Platoon level and constantly asked the Staff to pursue issues of no real value to be answered in the next briefing (two per day most days, 0800 and 1800; probably averaged only 1-1.5/day or so over the year, sometimes given to the Deputy Commander, the XO or, even the S3 whether necessary or not...).

So your comment:
This is related to the thread because the different command styles inevitably impacted the size, shape, and focus of the staff - both staffs were, by the way, simultaneously bloated, undermanned, and undermined by the need to continuously rotate in augmentees.
is quite accurate and, i'd add, affects units down the chain.

The only difference to that in my examples was the Staff for the first rather austere and distant but truly excellent Commander was fairly small but not undermanned while over the months that for the second cited which started as a small but fairly good crew working for a really nice but overly busy guy grew to be just what you describe due to a lot of make-work. Said 'augmentees' of course ripped off from the Battalions...

No doubt in my mind which of those Brigades functioned best and did the better job. Both were good, the first was far more combat effective with far less hassle; benefiting from a good example -- and, as you wisely illustrate, at least partly the impact of a Commander on the Staff.