Can, or do? How frequently? How much lower is their viral load, on average, than an unvaccinated person's?

You are either making a truly extraordinary claim, or are supporting my point. If vaccination significantly reduces viral load in 95% of cases, it's true that it might not reduce it for everyone. But it makes a huge difference, in terms of public health. Both epidemics, and viral infections are a numbers game. Reduce the denominator in an exponent, or a constant multiplier, and you get the difference between life and death.