ATCer here. I'm a Tower guy, so explaining what Center/Tracon dudes do can be difficult because 1) it wasn't me and 2) the equipment and rules they use are very different (even from each other). As absurd as this guy with the phone seems, keep in context the terrain surrounding the area, and the locations of radar antennae and radio transmitter/receivers - for Centers they're optimized for talking to people in the flight levels above 10k, and around terrain (which there's lots of in that area) you lose line-of-sight and radar contact/comms very quickly below those altitudes. Assuming the aircraft is semi-controllable, which it obviously was, and someone on a multi-crew aircraft is able to maintain communications, passing along a phone number while you can is a prudent move and it wasn't just so the mishap aircraft could cancel IFR on the ground. There's lots of info needed and it's likely that was rapidly going to be the only way to communicate with the Center, who can pass around useful stuff like Lat/Longs and other crucial info for emergency response and not just rely on eyewitnesses calling 911. Also do J-models have integrated Sat Phones, like the C-17's Aero-I?
Also where this occurred is just beyond the eastern fringes of the servicing approach, SoCal Tracon (SCT), and at the lower levels of Los Angeles Center (ZLA). SCT in the area is really set up to work the Palm Springs TRSA and to feed/sequence the satellite airports in the area (Bermuda Dunes and Thermal), and there's a basic ATC procedure of not forcing radio frequency changes on emergency aircraft unless better handling will result. Someone in an emergency descent from the low twenties or teens (not sure where MC-130s do their A/R) isn't a whole lot of time to work with beyond an emergency point out as you blow through someone else's airspace.
Finally the LiveATC tape was edited so not all the comms are there.
Anyway consider the limfacs, don't just look at it as quibbling. Very very happy there was such a happy end result. America!