This is just misinformation that’s been repeated so much as a meme people think it’s true based on a misreading of the simple fact that a study which said if you do experience a breakthrough infection with a high viral load, vaccinated people transmit that viral load as readily as a non-vaccinated person. What is clearly not true is that vaccinated people have the same likelihood of having an infection with that high viral load in the first place. Which they don’t and why vaccines remain an important tool to reduce spread.
In addition to averaging lower peak viral levels for infection, vaccinated people has faster clearance rates, which means they are contagious for shorter amounts of time: https://www.medrxiv.org/content/10.1101/2021.07.28.21261295v...
Vaccinated people have faster symptom onset as their bodies react to the virus faster, which means they can test and isolate sooner.
Vaccinated people are still less likely to be infected than non vaccinated people: https://spiral.imperial.ac.uk/handle/10044/1/90800
Vaccinated people are also less likely to collapse the local medical system in surges because their disease course is less severe: https://academic.oup.com/cid/article/73/12/2257/6124429