There are several points of confusion about what a Romanization is and what tradeoffs exist in this, and I’d like to address them.
The sort of consonant “p” and “b” here can be distinguished in two main ways: whether your vocal cords move and whether you expel air from your lungs. There’s a tendency—exceptions exist—for languages to hear one distinction as more important than the other. Most European languages distinguish “p” and “b” based on whether your vocal cords move; /p/ means they don’t and /b/ means they do. (I’m glossing over A LOT of detail here, but if you’re in a position to make a clarification about lenition, this post isn’t aimed at you!) Whether or not air is being expelled is generally heard secondarily, and while you’ll have a marked accent, if you get it wrong it’s much easier to understand than if you mix up the moving vocal cords.
Korean has the opposite approach where it first makes a distinction between whether a puff of air comes out or not and the vocal cords are a secondary consideration. A fun thing to do if you’re learning Korean is to light a candle and see if you can say “pa” near the flame without making it flicker. That is the sound /바/ makes in 바둑. European language speakers tend to hear this as a “p”, especially if they don’t aspirate the /p t k/ series; English does typically aspirate those letters and it’s a bit more of a mixed bag what we hear. However, in the middle of a word, 가방 (gabang), it is closer to a “b” sound like most European speakers would imagine.
Rather than trying to pick an intuition, MR does both. The words paduk (바둑) and p’aduk (파둑) are different, and you can see that difference unambiguously in the Romanization. The criticism has always been that it’s ugly and overkill, but RR requires you to fill in more of Korean’s phonology from knowledge of the language. Given some of the confusion on this thread about aspiration, I think that requirement is not intuitive for a lot of foreigners.
And far from being an “alien” phenomenon, this is roughly the difference between pi and beta in ancient Greek; admittedly not the Latin alphabet, but hardly a blip on the European linguistic canon either.
(Emphasis mine.) MR does not purport to transcribe hangul, and actually dates from a time when pure hangul was not the primary written form! Instead, it purports to transcribe some phonological facts that hangul requires you to assume. For this reason its still preferred by linguists working on Korean. The /p/ versus /b/ distinction it chose was the expected European one, and in that context /paduk/ is the more natural choice, far from it being arbitrary. The case for RR is that it is closer to a transliteration of hangul, though it makes some compromises there such as /seolnal/ versus /seollal/ for 설날. The bigger improvement is elimination of diacritics; no one has the keyboard to travel to Sŏul. The point being, /qa/ doesn’t map onto /바/ the way even /pa/ does.
At the bottom of all this is the fact that the Latin alphabet is a terrible fit for Korean, no matter how you split it. I would agree with the more measured point that “paduk” is old-fashioned, out of step with the South Korean government’s stated preferences for Romanization, and potentially misleading for those not familiar with Korean phonology if that’s what you mean to say. But it is not wrong because MR is a principled, established Romanization.