Or maybe SFO is already over capacity and can’t safely take the traffic assigned to it?

Then it's literally ATCs responsibility to inform the incoming aircraft of that. That's why there's a diversion airport.

The controller didn't have an ETA, the pilot was offered to hold or divert. The pilot didn't want to divert, and hoped that complaining would get them out of the hold.

From a safety perspective, this all seems to have worked as intended.

The problem is that the hold is effectively indefinite in the controller's mind, and therefore not an option.

But that's not what they said on the radio.

Delays are estimates. They can’t be guaranteed. All it takes is one other plane to be slow to act.

Estimates and predictions that turn out to be wrong are still wrong.

And when the estimate was wrong by 50% and counting, and the ATC wasn't offering any information other than another dubious estimate, and the ATC was not handling the flights in a FIFO order, the Lufthansa pilots were left with uncomfortably little useful information about their situation.

Just a note the estimate wasn't just wrong by 50% the pilots had already been told 10min twice before, so estimate was more than 300% off by that time.

When flying and being responsible for an aircraft full of people - estimates take a very different meaning.

That’s why flight plans have diversion fields, with requirements to have enough field to get to the diversion airport, and even to hold _there_.

That also means that when ATC tells someone they will inform them in 10 minutes,they are obliged to tell them in 10 minutes, even if it's "sorry, we don't have a slot, we can get you info in next ten minutes or help you to diversion airport".

Not have to be reminded that there's a plane in holding waiting for information.

Unless they want a repeat of telling a plane to hold till it crashed into sea.