There are many reasons why we might find major network radio shows more trustworthy. One reason is that, because of their size, they are likely to have more staff. A larger staff means that more stories can be covered, but also more fact-checking can occur. This makes the information more likely to be accurate.
Another similar reason is that major networks tend to have more money. With more funds, you can conduct more lengthy research, hire more capable people, and obtain access to topics and people that smaller networks would be incapable of accessing.
Finally, major networks have a reputation built over the years that they want to protect, and that could be damaged if it became obvious that their information is inaccurate.