Your conclusion is blatantly misinterpretation the statement.
An unvaccinated person who has previously had covid and an unvaccinated person who is still infected are two completely different groups of people.
Re-infection is very rare (but has been documented, regardless) among both the vaccinated and unvaccinated. So, the lack of data of those re-infected transmitting to others is not exactly proof that the unvaccinated do not spread covid.
If this were true, then there would've been no pandemic at all. And no, that is not proof there was no pandemic. Lack of data is not evidence nor does it discount the evidence that all unvaccinated people are capable of transmitting.
Only 0.23% of the US population has died from Covid. (Roughly 1 in 400 of the entire US population.)
Only 14.2% of the US population have actually been infected with Covid. (Roughly 1 in 7 of the entire US population.)
But 1.6% of that percentage have died from Covid. (Roughly 1 in 63 of those infected in the US have died.)
Only 58.7% of the US population have been vaccinated.
That rough 41% of unvaccinated who are among the 85.8% who've yet to be infected can still spread covid while infected (from the vaccinated and unvaccinated) and have about 1.6% chance (and potentially higher) of not surviving.