The only exception is for claims so dumb ("the time complexity of mergesort is O(n), the earth is flat, 1+1=5") that anything at all directly contradicts them.
Tertiary sources are often reporting on secondary sources: The NYT reports on something that WaPo reports on. Any mistakes or biases present in the secondary source may be amplified & added to. I never read reporting about someone else's reporting without seeking out that first reports. At best, tertiary sources should be taken as commentary/opinion/interpretation.
Non-primary sources are for when you want something explained to you in non-jargon terms or a TL;DR, or you want an expert's opinion on the primary sources etc. A medical study comes out about hear disease, and you want a cardiologist's assessment of the study. Even then, you need to understand who the cardiologist is. The average cardiologist may not be a great source: There are levels of expertise within any community, and I want someone who is at the top of the field. I don't want the average cardiologist's opinion.
The problem is that no one has time to track down primary sources for everything they read/see/hear, or the level of knowledge to evaluate it. I'm fascinated by quantum computing, but I have no chance of understanding primary source research there. I have no choice but to rely on a secondary source or, if possible, the researchers themselves giving a higher level TL;DR.
My rule of thumb is that, absent prior knowledge of bias/agenda/ill-intent, I skeptically take what people say at face value but don't accept it into my own views/opinions/etc unless I've done some digging of my own. If I'm not capable (e.g. quantum physics) of doing that, I simply try harder to get a variety of expert opinions, and avoid presenting anything I say/think as anything other than an outsider's imperfect understanding.