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.
You can catch flu from an asymptomatic person, but Covid has a much higher reproduction factor. During the winter lockdown in England, regular flu was completely eradicated - literally not a single case was detected in entire England [0]. At the same time, Covid was still spreading happily. The measures that stopped flu in its tracks only slightly inconvenienced SARS-Cov-2.
Covid is simply too good at spreading, compared to other similar diseases.
(As an analogy: I can swim, Michael Phelps can swim, we can both call ourselves swimmers, but we are not really comparable.)
[0]https://www.independent.co.uk/news/health/flu-cases-covid-en...