That nature article is still repeating arguments which conflate studies of Ct values of RNA with infectious virus and don't address the fact that vaccines reduce symptoms and that Ct values drop faster in vaccinated individuals.

The CDC article doesn't say anything other than "People who get vaccine breakthrough infections can be contagious" without any citation.

The actual study you cite only tests for viral RNA not infectious virus, they only cultured a single sample from a breakthrough infection to show that it could happen.

Other studies have found that viral RNA drops faster in vaccinated individuals, and that levels of culturable virus are lower at peak in vaccinated individuals indicating the presence of more viral debris than infectious virus:

https://www.medrxiv.org/content/10.1101/2021.07.28.21261295v...

https://www.medrxiv.org/content/10.1101/2021.08.20.21262158v...

We also know that viral loads are correlated with transmissibility and symptoms, and that vaccines reduce severe symptoms. We know there's less infectious virus in breakthrough infections and that they're less severe. It is rational to expect a reduction in transmissibility.

The point is; Is it more (/less/no change) transmittive vaccinated vs unvaccinated? If it lowers transmittion only a tiny bit it's still net positive.