If Bulow being a General Staff Officer was a factor then I imagine this institutional bias would come into play if the Germans decided to create a proper Heeresgruppe before the war then Bulow would still command it. The question is if an HG was created before the war, its command relationships and bigger 2-army mission spelled out would Bulow still micro-manage 1st Army, taking its Corps to assist Bulow own 2nd Army? Or would Bulow delegate (or be told to delegate) at least some command of 2nd Army to an ArmeeAblietung which would give him a bigger picture be default?
As Heeresgruppe commander Bulow should be thinking big picture but his control works down two levels, he needs to know what his two Armies and all his Corps are up to. What he should not be doing is issuing orders down to Corps, merely telling each Army the mission so they plan and execute. But he certainly could shift Corps, even Divisions, to reorient his force to the overall plan he is imposing as events unfold, he should be given the latitude, for most German command once the kick off is behind them they allow for much flexibility. The connections are from Chief of Staff to Chief of Staff, that odd to us Americans German back channel and "partner" in command. I am relying on it to mitigate the tendency to micromanage, if anything all these General staff officers will be very much in sync, if Bulow falls out of line or loses nerve those guys will challenge him or call for his replacement.
We saw this in East Prussia, Prittwitz panics, OHL is worried, his Chief of Staff gets jittery, Hoffman reports he has a plan but those two are faltering, OHL pulls them, finds Hindenberg (for backbone) and Ludendorff (for thinking) and Hoffmann's plan is found entirely as it should be by Ludendorff, a classic German solution to the problem, adopts it, endorses it, Hindenberg orders it done, oh Tannenberg, oh Tannenberg!
My other suspicion is that Moltke might have chosen Bulow because he was himself fearful of the big gamble he was taking and feared how aggressive Kluck was getting although not off the script. So an alternative here might be that Moltke chose Bulow because he wanted more caution. Here I can see him reigning in Kluck if the First strays too far from the objective, using his Corps as needed, or endorsing Kluck should events unfold to cause HG to see opportunity and confident in Kluck's staff's plans to do something off script. We might see the two Armies used to envelop instead of being held so close for example, I think that is why the HG level is necessary, it reorients the focus to one level higher vision and purpose. Something OHL could not properly manage on such delayed communications and information.