To hold Apple, or any other tech giant that happens to use tin-based solder (all of them) responsible for what's happening at the bottom here seems rather harsh. Why not critique the government itself?
You say that as if it's an either/or proposition. That's not how responsibility works. Let's walk up the chain again:
1. Some (relatively) poor people mine some tin without a license or whatever they need from the government. They are fully responsible for the legal, environmental, and ethical consequences of their actions.
2. The miners then sell the ore to a smelter. By accepting the ore in trade, the smelter is accepting responsibility for where it came from. This does not negate the miners' responsibility; it is a separate but equal responsibility.
3. The smelter then sells their tin via the state-owned exporter. At this point the state accepts responsibility.
4. Solder manufacturers buy the tin to make solder, adding themselves to the list of responsible parties. But their are many solder manufacturers, and they don't all use all of the tin, so their responsibility is proportional to the amount of tainted tin they take on.
5. Electronics manufacturers buy the solder; now they're sharing the responsibilty as well, again, proportional to their use.
6. Now we buy the electronics and must accept our share of the responsibility. But of the 55,000 metric tons of tin produced in Indonesia annually, each of us individually may be responsible for a few to a few hundred kilograms, depending on what we buy and what we make.
In short, responsibility is not like a hot potato to be passed around; it's a virus. Passing it on doesn't mean you're rid of it.
Looking down the chain, we each individually bear a small burden of responsibility. Apple, the solder manufacterers, and the Indonesian state can all be considered aggregators of responsibility, because they hold responsibility equal to that of all of their customers. From the state to the smelters and then to the miners, responsibility is spread back out a bit.
So from that point of view, the Indonesian state seems the most logical point of attack, because as you say its monopolistic position means it aggregates the sum total of responsibility for illicit tin mining in Indonesia. From a purely ethical standpoint, it is the largest holder of responsibility.
But from a strategic change management position, Indonesia is a very poor target, because it is a sovereign nation, giving it the explicit right to control both the definition and the enforcement of laws within its borders. If it decides to define laws such that these people are mining illegally, but not to enforce those laws, then it has that right. Certainly there are international treaties and human rights laws which can trump these national laws, but enforcing those is difficult and has unpredictable levels of success.
Apple, on the other hand, is an excellent target, for reasons which appear to be deliberately engineered by Apple. Apple has positioned itself in the high-end aspirational range of the electronics goods market, partly by setting very clear social responsibility goals for itself and its suppliers, and by enforcing them.
So if you actually want to effect such change in an arena where Apple is a serious player, you highlight Apple's responsibility, because Apple's position is sensitive to such criticism.