Is this actually true? It is certainly not true for HIV, and of course is not relevant to diseases like Zika that are transmitted by mosquitos.
Edit: I found the answer to my own question: https://www.kff.org/infographic/ebola-characteristics-and-co... (see second bullet point). Given that this lists Hep C, HIV, Influenza, Malaria, Polio, and Tuberculosis as possible to transmit while asymptomatic, I'd say "COVID-19 is one of the few serious diseases that can transmit when the carrier is asymptomatic." is most definitely false.