The question gives a number of possible advantages, so I will post possible disadvantages. It is then up to the scientist to evaluate the tradeoff between possible advantages and disadvantages.

1. The reference makes it sound like there are considerable computational advantages. Especially in deep learning settings, the computational issues really do have to be considered. While it is great to be able to prove, mathematically/statistically, that some method is superior to a “trick” in deep learning, if that method cannot be computed in a reasonable amount of time, it is not useful. However, I am not sold on the computational advantages. For instance, if the classifier makes a high-confidence prediction that an observation right near one of the bin boundaries is in the bin on the other side of the boundary (probably only a mild mistake), the entire loss could be dominated by that when you start taking logarithms of small numbers in the cross-entropy loss. This puts the model in a position to get hung up on fixing a mild mistake, perhaps at the expense of making improvements in other areas where the errors are more egregious.

2. The fact that a classifier returns the probability of being in a particular category is appealing. However, neural networks are known to be overconfident in their predictions of these probabilities, and calibrating a multi-class output is not straightforward. Further, techniques exist to estimate conditional distributions, such as quantile estimation.

3. There is not an especially high penalty for bad misses. If the prediction puts high probability on the next bin over from where the observed value is, that incurs the same penalty as putting that same probability in a much higher or lower bin. While this could be argued to give robustness similar to how minimizing absolute loss gives robustness in that large misses are not penalized as severely as they are for square loss (for better or for worse), at least absolute loss penalizes more for large misses than small misses. There is a limit to how much robustness is desired.

(The first and third disadvantages can be combined to say that this approach risks giving large penalties to small misses and small penalties to large misses.)

4. Some of the appeal of this seems to come from classification accuracy being easier to interpret than regression metrics like (root) mean squared error. However, people goof up in interpreting accuracy all the time. I cited a paper on here a few weeks ago (Sundaram & Yermack (2007)) that seemed to be raving about achieving a classification accuracy of $97\%$, despite the majority class making up $97.71\%$ of the observations, meaning that a naïve model could achieve $97.71\%$ classification accuracy (better than their model achieves) by predicting the majority category every time. (This article was published in the top journal in its field (not “a” top journal, “the” top journal), so it is not just the fringe that makes mistakes in evaluating classification accuracy.) Even when the Sundaram & Yermack (2007) classification accuracy scores are above the scores achieved by predicting the majority category every time, the reductions in error rates, which is probably more informative (and can be equivalent to Cohen's kappa), does not scream out, "This model gets an $\text{A}$," the way that a classification accuracy of $97\%$ might. Further, regression metrics like root mean squared error and mean absolute error are in the original units of your measured outcomes, which should have an interpretation by someone who knows the field.

This answer to "Why should binning be avoided at all costs?" is worth a read, even if it is not about the exact same topic. I especially like the last sentence, which I will quote below

My recommendation would be to learn the analytical methods that are applied to the underlying continuous data, and then you will be in a position to determine whether a crude approximation via binning is necessary in a given situation.

REFERENCE

Sundaram, Rangarajan K., and David L. Yermack. "Pay me later: Inside debt and its role in managerial compensation." The Journal of Finance 62.4 (2007): 1551-1588.