In its specifics, dispatching and the dispatch environment varies widely by time and place, but the basic "job" and skill sets required are pretty universal - multitasking, prioritizing, detailed long- and short-term memory, maintaining calm and control within yourself and the people you're working with (fellow employees and the public), information evaluation and ordering, etc.
Working at a big Dept like LAPD may have its challenges... but I would challenge any of them to 'do it all on their own' like we do at smaller municipalities.
I took that challenge years ago when I began dispatching for Grass Valley PD (11 officers, 4 dispatcher-clerks at the time) after having previously dispatched for LAPD and supervised at South Bay Communications, then in Redondo Beach, and had no problems. After a couple weeks of training I went solo on the one-person combination dispatch center, records division, and front desk.
As I'm sure I've mentioned here before, my absolutely busiest day dispatching ANYWHERE was one Sunday there when a freak windstorm blew in, toppling trees, power and phone lines, and all the rest. While my couple of officers and one sergeant were running around like crazy checking and assessing the damage in an attempt to sort of triage everything, I was also the weekend Public Works dispatcher and FD's "tactical" dispatcher. Half the town's 6000 people were calling in on my two 9-1-1 lines and four business lines to report problems or to ask "when will it be fixed?" <sigh - like I'm gonna know?>, and it seemed the other half who'd lost phone service were lined up at the front desk for the same reasons. And PG&E kept calling asking for traffic control here, barricades there, "we've got wires down in the street what's the officer's ETA?" Right. :roll:
It wasn't on the same life-safety scale as handling 3 or 4 simultaneous shootings and 211s plus the routine calls in 77th Division on a Summer Saturday night, but as you allude to, in this case I had to do everything literally by myself. Couldn't even reach any of the off-duty dispatchers for help, and I'd only been there perhaps two months at the most. But the identical skills and habits I had (or had developed) kicked in and we all got through the day not too much the worse for wear. I was never "super-dispatcher" by any means, though my ratings were always above average, but as you're well aware, it's the very nature of the job to constantly be expecting and reacting effectively to the unexpected, and coping with new situations that for a quick second may seem filled with copelessness.
Over the years a number of my LAPD co-workers moved off to smaller agencies - South Bay, Huntington Beach, San Marino, OCFA, CSU-Northridge PD, Newport Beach (or maybe it was Costa Mesa), some town in PA, another one to a little burg in MA, etc, and as far as I know they were all able to cope with the change and stayed until retirement or are still on the job.