Your reply equally skims the surface as the article. Given below are the excerpts from the article you shared:

Point 1: "Reports suggest that unvaccinated people infected with the delta variant have a roughly 1,000-fold higher viral load than with the original strain of SARS-CoV-2"

It compares between two variants (alpha and delta) in lab model and not in real world.

Point2: "suggesting that the viral load in breakthrough cases of delta isn’t just high. It’s equal to the loads seen in cases among unvaccinated people. What does the evidence show?

This claim has caused a lot of concern in recent weeks because it suggests that the protection offered by COVID-19 vaccines is eroding. It’s primarily based on a CDC report that described an outbreak in Barnstable County, Massachusetts. In that case, researchers tested infected individuals for a parameter called CT, which indicates the presence of viral RNA and is often taken as a proxy for viral load."

As per RT PCR test in real world both vaccinated and unvaccinated have same viral load for delta variant according to CDC. Given below is also a Singapore study based on real world which found similar viral load among vaccinated and unvaccinated

https://www.medrxiv.org/content/10.1101/2021.07.28.21261295v1