The statement is true. If you're not vaccinated, and you get COVID, there's a non-zero chance that you'll be the source of someone else's breakthrough case.
This statement is also true: If you're vaccinated, and you get COVID, there's a non-zero chance that you'll be the source of someone else's breakthrough case.
So, the quote is kind of useless. The unvaccinated probably do spread COVID faster and with more frequency. So, the directionality of the quote is fine. But it reads like a boolean statement, which is incorrect. COVID is almost certainly endemic at this point. Vaccines clearly help, but it's incorrect to blame all of the mutations and spread on the unvaccinated.