There are a lot of factors. Outside the DRC cobalt is almost always a byproduct of nickel mining- Due to nickel supplies coming back under control (the price fell 80% in the past 10 years), cobalt prices haven't risen much. The main thing is that a LOT of new mines have opened/will open. Cobalt hadn't been mined in the US for over 30 years until now. We had no nickel mines until just a couple years ago. Now I think there are two cobalt mines opening and six nickel mines, in the US alone. In Canada and Australia even more have been opening. Tesla in particular has been very canny about securing contracts (Indonesia iirc? Plus several in the US... once they open). They've been publicly discussing cobalt for the better part of the decade and spurred a lot of investment in mines, and are very well prepared.
Note though that you're right: there is some potential for the short term (~3 years) price of cobalt to swing. Batteries make up a solid ~50% of global cobalt consumption. That's almost all due to China- in the US almost 70% of cobalt is used for metal-related stuff, mostly superalloys for turbines and jet engines. It's basically a race between new mines being able to ramp up production and the battery revolution taking off. Tesla isn't that worried because while cobalt is important, they can bear a short term cost increase. Tesla has repeatedly said they're under $200/kWh, and even at that price the battery would be $10,000 on a model 3. Of that the current cost of cobalt is probably ~<5%- if the price of cobalt quadruples it's still a total of <$2000. That's almost 6% of the cost- painful, but survivable. And that's an extreme situation.
Another reason they may seem blase is that cobalt is far from the biggest single cost in the battery. It's just the most potentially volatile. Only about 9% of the cathode is cobalt (in NCA), and nickel and even graphite are much bigger costs. Those two are significantly more stable- graphite can be made fully synthetically in a pinch (currently ~45% of battery graphite is synthetic[1]) and nickel consumption is already so huge (because of stainless steel) that even if every car was electric, it would hardly affect our nickel consumption. Cobalt prices can go up quite a lot before it actually affects Tesla's bottom line, but nickel and graphite will affect it immediately. In the end they all roughly balance out.
[1] more info: Synthetic graphite is made from high quality anthracite coal or refinery tar (asphalt, bitumen) that is heated in an oven (carbeurized, then graphitized) and then ground in a ball mill (then chemically treated, plus a whole bunch of proprietary voodoo). It's roughly the same cost as natural spheroidal graphite (the kind used in batteries- think best of the best of the best) and even cheaper sometimes, but they have different qualities. Natural graphite has a slightly higher capacity, while synthetic graphite has higher power capacity, so they use both in a blend. 90% of making batteries is picking the right blend for your application- there are hundreds of every ingredient to mix, tweak and balance. Competition is achingly incremental and fierce.