The BAS as a whole is an outgrowth of the "Scientists' Movement" of 1945-1946 or so, which involved former Manhattan Project scientists (and others) working with lobbying and advertising firms to find ways to amplify their views on policy. The Clock is an instrument of that goal, as is the BAS as a whole. The motivation for doing these things is that said scientists felt that they had a special expertise to offer the country and the world, one that was not being adequately represented by the military or the government, and one that was not adequately represented by individual scientists who were, on the whole, not involved in making public policy pronouncements (or whose individual pronouncements could be dismissed readily as individual opinions). The tone and politics of the BAS have shifted over the years, as have their issues (they have expanded beyond the narrow frame of nuclear war, though it looms high in their minds). For more on the Scientists' Movement, Alice Kimball Smith's A Peril and a Hope is a great reference to start with, and discusses the origins of the BAS, FAS, etc.

On your more soap-boxy part:

Considering it is run by the Bulletin of the Atomic Scientists, who no disrespect, are not international affairs scholars and this not really qualified to make such judgements.

The clock determination is made by the BAS's Science and Security Board. It includes a wide variety of experts, including but not limited to international affairs scholars.

It seems to be given legitimacy and is constantly written about when updated (as happened today), though there is no apparent basis for how they reach these numbers, no defined system aside from their own beliefs.

The clock setting has been, in recent decades, always accompanied by a quite lengthy explanation as to why the board changed it. "These numbers" seems to imply that you are mistaken about what the clock is meant to represent — it is a means of summarizing the opinion of a group of experts, not a scientific dial. It has never self-represented as the latter. Its setting is not the result of an equation or "data."

While I could be totally incorrect here, it seems like it's just an opinion piece that has been given undue weight for whatever reason.

It has been given weight (undue or not) because, historically, the BAS has been composed of many leading national and even international experts on the subject of nuclear war (and now other topics, like climate change), and the collective opinion of a bunch of experts is, arguably, worth something, even if it is, alas, "an opinion piece."

If you are asking, why do news agencies report on this? The answer is somewhat tautological: because they think people will be interested in the story. If you are asking, why do people care when a bunch of nuclear weapons experts make an announcement that they think the trends of the world are leading to disaster? Again, refer to the Cold War context of its origin, and, if you want to be more present-day about it, consider the present national moods.