I think we need to ask ourselves: when we seek out the news source's agenda first, what are we really trying to accomplish? Are we trying to think critically about what we're about to read? Or are we applying a quick heuristic filter to decide whether to read at all?
Think about it this way: I come across an article on XYZ site. I determine (or guess) that XYZ's agenda is generally conservative, and I consider myself a liberal. What do I do with that assessment? Do I stop reading and move on? Do I preemptively and summarily discount anything the article has to say? Or do I continue to read, perhaps skeptically, but nevertheless with an open mind?
I'd like to think we're doing the latter. But I'm not convinced.
My point is, it's totally fine to consider the source when making assessments of an article. It's totally fine to consider the source's agenda. But if seeking out the agenda becomes a routine, primary step in the process of consuming news, that worries me. That's essentially just seeking out confirmation of one's own biases as a reader.