What makes Tesla unique in this respect, and Musk so essential to the operations?
I'd argue nothing at all, other than his image. Hire some experts from the other companies, people with a few decades of experience, and let them turn it around. Maybe start with why you're hand-assembling and gluing things together in tents, rather than using the high efficiency production lines everyone else is using.
Elon is also an engineer first, that is needed at innovative companies, not some metric focused MBA C-level executive just looking to cash out. Musk actually cares about what he is building and has done a smashing job considering the pressure in my opinion. Tesla is made with love, a car/product people want, not just for the short term numbers.
Your comments regarding being "made with love" doesn't square with all the other articles I've read recently regarding the poor fit and finish of new Teslas. They are having problems scaling up, and it looks like quality took a hit as a result.
Running a large scale industrial manufacturing operation means extensive delegation. Those "metric-focussed" executives actually have an important role to play at this scale. Product design and engineering happen well before scaling up for production. The production side is all about supply chains, efficient assembly and QA. No one should be changing the design at that point; however talented Musk might be as an engineer, I can't see him specifically having much to bring to the table for sorting out manufacturing issues. (I have worked doing QA in an industrial setting. ISO 9001, six-sigma and all the rest do start to have a point when you are working at scale and want to guarantee quality, throughput, efficiency, customer deadlines etc.) Tesla is bringing nothing new to the table on these fronts that I have seen.
You do need a focused innovative leader to run innovative companies that are competing with big players, if you get a metrics guy as the CEO or one calling the shots they will fade away.
Taking away the founder/creator is like how Python is without Guido and Linux kernel without Linus, they will eventually run out of momentum and get convoluted with many directions trying to take the lead for short term needs, similar to how node went for a long time or Microsoft under Ballmer or Apple under John Sculley (they thought an exec guy from Pepsi could run a technology company).
The typical process only focused metric MBA guys should come in when the innovating is done and it is time to focus on short term revenues over product, which eventually runs out everytime as it is milked and research and development is killed.
There is a reason Amazon is so successful still, Bezos is an engineer/product guy first that is laser focused on taking profits and putting them back into R&D, hard to beat that as you can see.
This has little to do with producing cars, and more to do with (financial) management. (Also, most major european car makers survived the great recession just fine).
Most of the European (Mercedes/BMW) engineering and product people have power. Even Honda for a long time engineers/product people drove most of the decisions and power structures. American car companies not so much, most MBA / bizdev lead over engineering and product people. Germany is unique in that engineering and the product quality is highly respected over short term gains only.
Additionally, BMW/Mercedes/etc cars also target the upper classes that weren't as affected by the Great Recession so they didn't have the pressures as much as the US car companies that needed bailouts. American industries targeting consumers in upper class and wealth didn't suffer as much.
That's pretty much any company (with certain variations for different industries).
What about when Elon was spreading rumours that the Saudi's wanted to take Tesla private?
Though internet companies aren't the best at cash flow management either, it is more about getting the market / customers then the profit. I think innovation is excusable for revenue/profits for some time.
Starting an electric car ONLY car company is even harder than having it as part of an existing car company like Mercedes is doing well with.
The shorts are mainly trying to kill Tesla and coming from the oil, gas, other auto companies most of all. Lots of games are played with car companies (see VW short squeeze a historically big one) because they can't react as fast as it is hard work. I think the shorts have been not just focused on profitability but to drive out the competitive threat, the shorts have also been attacking Elon Musk non-stop.