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.
They sampled people who were getting tested for COVID, and divided the samples into vaccinated-with-breakthrough-cases-who-got-tested and unvaccinated-and-sick-who-got-tested.
This completely fails to account for people who are vaccinated, and don't have breakthrough cases, which is the overwhelming majority of vaccinated individuals.
Because of the incredible sampling bias in the study [1], the conclusions that it makes are significantly less extraordinary than your claims.
Just because someone somewhere runs a study that compares 50 dead-in-car-crash people who were wearing seat belts, and 50 dead people who weren't wearing seat belts, you can't conclude that wearing a seatbelt does not reduce your risk of death or injury.