You're presupposing that most people base their opinions rationally on facts, and that it's easy for people to agree on whether a fact is true and relevant. Neither is remotely true.

Every individual is swimming in data from the moment they're born. By the time we're adults, we have thousands of interrelated beliefs and opinions, based on personal experience, anecdotal reports, and things we hear and read. The strength of an opinion is directly affected by how much we value and admire the source. Not all sources are equal, and we don't treat them that way.

I, too, notice the "growing disconnect between facts, opinions, and beliefs". I attribute this to the growing cacophony of ideas being presented, the number of different voices and the breadth of the viewpoints they express, coupled with the lack of true authority. Respected journalists used to make an effort to be objective and get the facts right, but in the last ten years (and especially the last 5 years) all pretense of objectivity has disappeared. It's gotten to the point that they can't even be bothered to check for typographical errors.

So then, if I hold an opinion because it came from someone I respect, and my experience confirms it, but the New York Times reports an apparent fact that directly conflicts with my opinion, what to do? Could it be a mere factoid, a claim that looks like a fact in its presentation but that falls apart when you read the journal paper the factoid is based on? Unfortunately the journal paper (or defense department report, or interview notes) is rarely available.