The arguments in the video are mostly misdirected, conflations of related but distinct issues, or (hypocritically) just opinions. Here are the unpacked issues as I see it:
1. Whether to summarize data by a handful of summary statistics or a full density. Obviously, some statistics reported in isolation can misrepresent the underlying distribution, but these considerations ultimately depend on what specific point one seeks to make with a plot. There's no reason a priori that visually annotating summaries/quantiles on a distribution plot can't be helpful (quite the contrary).
2. Whether to "smooth the data" (read: perform kernel density estimation). In some sense this is a long-solved problem: there are mathematically grounded methods for estimating the optimal KDE bandwidth (with varying degrees of assumption on the underlying distribution), which are what's used by any serious plotting library. And whether authors adequately describe what they're actually plotting is a separate matter. That said, there are many reasons not to show a KDE over, say, a binned histogram, especially with raw data and/pr small sample sizes, but these are entirely orthogonal to the choice of displaying a KDE as a violin plot versus something else.
3. How to normalize densities. With raw data, you probably want to compare frequencies (a proper pdf). If displaying just a single distribution, there's obviously no reason not to show the density (it's only a trivial rescaling of the axis tick values). When comparing multiple, the decision again depends on the point of the plot. Losing dynamic range for broader densities when compared with narrower ones can be counterproductive. E.g., in Bayesian parameter inference (where the data are MCMC samples), we almost never compute the actual normalization factor, but rather want to compare relative probabilities (i.e., within a single distribution) of different parameter values across different distributions. Of course, nothing forces one normalization over another for violin plots.
All of those are separate (and rectifiable!) issues from the defining characteristics of a violin plot:
1. Distributions displayed vertically rather than horizontally (both being harder to interpret and inappropriately suggestive). We almost exclusively visualize functions plotted vertically across a horizontal coordinate. I think this is the only valid, specific criticism of the common violin style itself, but the fix is of course trivial.
2. Horizontal violin are then only different from a ridge plot by a) not overlapping (which to me is a major improvement over standard ridge plots, but also trivially fixed) and b) being displayed symmetrically. I find it slightly easier to compare relative heights in the symmetric version, especially when comparing many distributions (such that each is relatively narrow). Even if not, the difference is so superficial/trivial that I don't find it worth arguing about.
Beyond this, the video's main argument (repeated every minute) seems to be that "it's bad, it's just bad", but there are only so many ways to make a 5 minute argument fill a 42 minute video. (This style of video is so grating to me.)