There's a grain of truth to the idea that Columbia fills its spots earlier than other programs, but I think it's going way too far to say that you shouldn't apply to Columbia during their equivalent of round 2.
Columbia's "binding" early decision program is basically a yield management tool for them. An overwhelming majority of early decision admits will ultimately choose to attend Columbia, partly because they've agreed to pay a non-refundable $6000 deposit if they're accepted. Because of the early decision round, isn't a surprise that Columbia's yield (the percentage of accepted students who choose to attend Columbia) is a little bit higher than other schools in its weight class:
https://poetsandquants.com/2012/01/17/th ... n-rates/2/But I don't think that Columbia's admissions standards are all that different from those of other programs. Just like any other MBA program, Columbia does its absolute best to attract the most brilliant MBA students it can possibly find. The adcom isn't going to admit crappy students just because they applied early decision, and they aren't going to reject strong applicants in January. Sure, early decision gives you a slight advantage if you're otherwise a borderline candidate, but Columbia isn't completely overstuffing its class in the fall, only to be left with no spots in January.
If you're a strong candidate, you'll still have a perfectly good shot at Columbia if you apply with the rest of the early January herd. It could be true that applying toward the end of January (several weeks after other b-schools' round 2 deadlines) hurts your chances somewhat, but even Columbia accepts a handful of students in March and April every year. So it's not all that different than anywhere else: if you're awesome, you'll get in during Columbia's rough equivalent of round 2. And if you're not awesome, you still won't get in during the early decision round.