First of all, there are plenty of conspiracy theorists with everything around us. Someone is going to come up with a video to make you believe in their snake oil. Do these 'doctors' have any factual data or research to share that suggests that these vaccines are potentially dangerous? I would actually love to know because I respect the facts.
Let's assume these vaccines carry risks, and these doctors are right. What are the alternatives? Every anti-vaxxer I have come across is also anti-masks, so there goes that protection too. Should we stay unprotected? The stats tell us 400,000 annual excess deaths in the country and about 30% average incidence of long-haul symptoms. We prefer this instead of a vaccine that has a who-knows-in-a-million chance of who-knows-what happening?
Before the vaccination efforts, the US was hitting almost 4,500 COVID deaths per day. Four months on, the daily deaths are down to just 800. In just these 120 days, the vaccines have saved 440,000 US lives! And probably millions of long haulers. So, no, that argument is not being silenced - it just becomes irrelevant if we won't be alive or healthy enough to talk about it.