As you suppose, it's the sourcing of the tritium that's the problem, but I think you're underselling the difficulty. For the DT fusion, the plasma composition can be mostly D, which is difficult but not impossible to purify out of naturally occurring water (prevalence is generally about 1 in 10,000 to 100,000 hydrogen atoms). Tritium is about 1 to 10^18th hydrogen atoms, which is a million million times less common than even the uncommon deuterium. Which means tritium needs to be manufactured, and at a certain point, the amount of energy being put in to make the tritium fuel will become prohibitive, making the economics of a T-T rector nonviable.