Quote from the study you linked: "[...] we found that BTIs[breakthrough infections] in recently fully vaccinated individuals have lower viral loads than infections in unvaccinated individuals. However, this effect starts to decline 2 months after vaccination and ultimately vanishes 6 months or longer after vaccination."
A few citations on why the evidence leans towards reducing the spread being more effective at reducing variant selection: https://twitter.com/trvrb/status/1462816243245719558?t=9OCuB... Here is a citation regarding viral load. https://www.nature.com/articles/s41591-021-01575-4