Heading out of the 28s the problem is San Bruno Mountain, with an elevation of 1319 ft. at Radio Peak. So add those antennae on there, and you get a MVA of 2600. In 1999, a United 747 mishandled an engine out and cleared the ridge by ~100 ft. In the evenings, the fog rolls in off the sea, and spills over the San Bruno range. In the mornings, it slowly burns off. I'm not sure that's the case here, but it would fit the clouds.
That's why the initial request to level out at 2000 feet was assigned 2600.
It was a busy frequency, so the first emergency call got stepped on (at least partially). On the second call, they asked for an immediate return, and NORCAL's concern was that, if they couldn't maintain their own terrain clearance, they climb to 2600. The first job of everyone, even above returning to the field, was to avoid hitting anything. If they can't see out the windows, NORCAL would really like them at an altitude where they can guide them via radar.
The guys in the Citation are having a rough time of it -- losing a door can and has finished badly --, but the yelling is probably due to the ambient noise more than anything else. With the windows down and the pedal to the metal, it's going to be really loud in there.
"We're having a hard time climbing, can you just send us back?" could be interpreted in a variety of ways. The controller explained why she was making them climb, and they clarified that they were unable to climb ("we're trying to climb – it's not doing well").
At that point, she had everything she needed, and, with coordination, turned them back below the MVA. 100 was the reciprocal heading, 90 was to intercept the approach for RW 10, then 70. They wanted an ILS approach to 10L/R, and she tried to give it. It took thirty seconds for them to figure out ILS 10L/R didn't exist (My guess: a combination of San Bruno Mountain and noise abatement means that landings are conducted on 28L/R most of the time, and 19L/R when the wind is weird). About 4 nm from the field, they enter VMC conditions and do a visual on 28R into a 10-kt headwind.
VX 904 could wait: they were climbing to 19,000 feet on the way to Vegas. Plenty of time to deal with them.
Sure, if you want to nitpick, there's plenty to nitpick. The flight crew does not act as if they have any idea of the terrain below them. Even before the emergency, they ask to level off at below MVA. That morning, they probably saw a wall of fog to the West and didn't think there was a mountain there.
"Uniform Pop"was not clear in their radio transmissions. Standard phraseology or not, it would have saved a lot of time if, when presented with minima, they stated they were unable to maintain them, rather than understating the case from inside a tornado of paperwork and screaming businesspeople.
Certainly, it would have helped if the departure controller knew that there were no ILS approaches to 10L/R, but she had already cleared the traffic behind them, put SFO on alert, and sent the Citation where it needed to go. She also could have pointed out the high ground below them using non-standard phraseology, since they also did not seem to grasp what MVA meant.