First off, I go with the Mechaber like many others.
Second, the Rambam holds that even cleaning your teeth within 6 hours doesn’t help. That even after cleaning one’s teeth he is choshesh for meat to be stuck in them. It is not a stretch to say that without cleaning ones teeth it is a vadai, that meat is be stuck inside. This would explain the chumra of chicken.
Thirdly, you have Rashi, who holds that the shuman stays in the mouth and gives flavor for 6 hours (aside for the problem of meat inside the teeth, which he holds is never batel, unlike the Rambam). I don’t see how you can differentiate between eating it together and eating meat and milk one after another, as we all know, ta’am ke’ikar. You may want to learn the sugya inside.
Fourthly, being that nowadays the general population (re:not everyone) accepted the halacha of the Rambam and the Rashi, an ashkenazi that is batel milk to drink after meat, is being mevatel “issur” lekatchila.