it was up to lufthansa to tell approach that they're not going to be able to accept the approach in use. It's being advertised on the atis that they would've been able to pick up in the air long before they got near the airport. (well, 10+ minutes before this interaction started anyway, they can probably get d-atis anywhere).
i'm sure it was said in the follow up video, but lufthansa knew what they were offering at SFO and if they needed something different they should've spoken up much sooner.
Is it really just a matter of taking off and hoping for the best, or did somebody in the chain of granting a take-off clearance miss something?
I wonder if an updated (international) flight plan is checked for projected congestion levels (based on other flight plans?) at all and can be declined for that reason?