While that's galling to see, it's targeted to convince the anti-vaxxer, or really the person who's on the fence. If they just dismiss the ideas, the anti-vaxxer says, "Oh look, the mainstream media covering up the truth again. Must be paid off by ${EVIL_BILLIONAIRE}." If he's given a chance, debates, and loses, that's a much better impression.

What's with this tendency to stick a binary label on every controversial issue? There are peer-reviewed valid concerns about these vaccines. Are the authors lumped into the anti-vaxxers category?

Regularly watching BBC World I've yet to see the both-sideism you describe. As if on the same level? Well in relation to another bete noir, you'll likely get to see an interview with Greta T. (zero academic qualifications) while waiting interminably to hear any (usually edited) counter-position from (for instance, there are many others) Richard Lindzen (with around 200 peer-reviewed papers in atmospheric climatology, MIT emeritus professor) or Judith Curry (https://curry.eas.gatech.edu/currycv.html).

That's arguably not "both-sides-ism" so much as "straw-man-ism" if one side does not get the best representation.

That said, it depends on the demographic you want to represent.