Yes. If I read the article correctly, the data in fact say you cannot determine whether the death rate in vaccinated people who become infected is different than in unvaccinated people.

However, 1) it isn't clear if Frieden is working from the same dataset as the author, and 2) a very slight re-wording of the initial statement would make it correct, i.e. "If you are _exposed_ to covid and you've been vaccinated you're about 100 times less likely to die."