Trust

I would say yes, because "doctors" are not necessarily experts on virology (Appeal to False Authority ). Further, advice on the Coronavirus generally incorporates other issues of personal/economic well being, which could be an issue depending on what is meant by "trust."

1) Researchers specializing in the Coronavirus and virologists are generally the experts here; doctors (as in generic practitioners) generally follow the advice of the specialists if they are good, otherwise they may be giving their personal opinion. No matter who the experts are, we should put our trust in the science. Doctors can ignore/be unaware of the current science.

2) When doctors give advice, they should do so with the understanding that there are choices, and make those choices clear. For example, "If you do decide to go out to eat, realize that you are increasing your risk of exposure to the virus." If by "trust" we mean "I acknowledge this, and I accept the risk," then I see no problem. However, if by "trust" we mean "This guy doesn't know what the hell he is talking about," but when the same doctor says "take two aspirin and call me in the morning" and the patient does so acknowledging the sound advice, then there is clearly an inconsistency in reasoning.

I don't know about other countries but in the UK when we talk about going to a doctor we mean a GP (general practitioner) who has a broad range of non specialised medical training. They deal with minor issues which are easily diagnosed and treated, and for more serious or difficult to diagnose conditions they refer you to a specialist. 

So that type of doctor I would trust to deal with minor things and wouldn't expect to be giving advice about managing a virus, something which is done at a societal as well as personal level. The type of person who should be giving such advice should be selected by the government (assuming you don't have a clown in office), and that person would take advice from doctors with relevant specialism rather than be a doctor. 

With regards to doctors in a America, my knowledge may be clouded by innacurate caricatures, but the impression that I have is that they often are under the influence of sales staff who offer incentives to sell their particular product, and indeed are driven by making money rather than proper treatment. This view is backed up by the opioid problem which is extant in the US, caused by doctors handing out oxy this and that to patients without any stringent control and after care, or indeed oft times without any need, and then dumping them when the governing body cracked down on them, leaving addicts to buy alternatives on the black market. Would you trust that type of person?

The question perhaps isn't should you trust a doctor based on one specific example, but rather should you trust them at all. I say that with the caveat that you should judge each on their own merits and not generalise (and perhaps the only thing you should generalise about is not generalising about evaluating groups of people).

Is it a fallacious argument? I think it's the wrong argument, and as such I'm not able to critique whether it's fallacious.