Intending that another interruption (from SLF/attorney) of discussion by the pros will be tolerated, maybe even interesting.
How does the aftermath of this particular situation get worked out, if it ever does get worked out? Intuitively and by common sense an outside observer would think within FAA, and also up the reporting chain of the established international air carrier, reports filed by the participants (or supervisors) will express frustration that such a situation could occur. They'll cite different reasons depending on whose report it is, and neither ATC nor LH will think another instance of this sequence of events is one that should be just shrugged off, not anything worth doing something to solve or prevent. But my uneducated guess is, FAA has imposed certain rules, and if I get the flow of the discussion FAA would back the ATC actions (because it sees inconveniencing one flight operator as better than disrupting a lot more other operators). I won't guess about whether LH's SOP makes sense enough to warrant its imposition (SLF . . .).
But ATC Watcher points to ICAO as intended to create - though the post didn't use these exact words - complete uniformity, even on the level of operational procedures for approach and landing. That's an overstatement, though - ICAO was created and organized, and today it operates, to create and uphold standardization. The fact that "Differences" can be filed establishes that uniformity of procedural choices for operational matters wasn't and isn't what ICAO does. (During her stint as Secretary General of ICAO, Fang Liu repeatedly emphasized in remarks delivered to professional conferences that standardization, and not necessarily uniformity, is the objective.)
If the FAA's view of the operational procedures the ATCOs were following is incorrect, who is going to direct FAA to change things? - you would think LH would be entitled to some positive action about this type of situation. Germany hosted the (very large) AirDefender 2023 NATO aviation exercise last summer, and the EUROCONTROL Global Civil-Military Aviation Summit was held in Brussels during that exercise. One would think that, if LH deserves some positive action to prevent a recurrence, somebody in the vast U.S. Fed. Government would motivate change. (I'm not referring to ATCO attitude or manners, just the facts or operating procedures.) And this even despite the more than a little inconvenient fact that United States at present does not have a Permanent Rep at ICAO and there appears zero prospect of the current administration presenting a nominee to the Senate.