I see this as a reap what you sow moment for Lufthasa. Why should ATC at a busy airport be so accommodating? Lufthasa is the one making this hard on everyone. I used to watch both these airports fairly frequently from Oyster bay regional park, they are both super busy with flights often lining up to the horizon.
It is literally ATC's job to facilitate the safe separation of aircraft. Note I said facilitate, not ensure, because ensuring safe separation and operation remains in the cockpit. When a pilot arrives at an airport and requests a specific approach, whether that reason is company policy or the limitations of weather, it is ATC's job to accept that request or deny it, and not to beat around the bush suggesting doing one thing and calling it another. When they give an expected time for something the pilot makes decisions in the cockpit if that new limitation will work with whatever limitations already exist. If ATC is not operating honestly then that should be viewed as what it is - a compromise of safety, and a petty unnecessary one at that. If ATC is unable to accommodate the request then it needs to be stated so explicitly and as soon as possible because lives are literally part of the equation. Air traffic can be lined up from SFO all the way back to London and that still doesn't change ATC's responsibilities one bit. ATC does not "accommodate" because that implies they exercise some arbitrary discretion and not clear binary criteria.