There is naturally a large literature on outliers and many threads here. It is not especially cynical, I hope, to give a one sentence summary that if outliers are not obvious (as I consider they are in the question) then their identification is necessarily arbitrary. The use of box plots to make decisions about outliers goes far beyond what John W. Tukey intended in the 1970s when proposing his particular conventions for what should be shown.
More generally, what is an outlier depends crucially on what is expected. A common pattern in my experience is that apparent outliers are not outliers at all -- if considered on a scale and with respect to a distribution that makes sense for the problem.
The question of visualization is interesting and important. In principle, the fraction of outliers -- which appear sufficiently numerous to be regarded as a distinct subset, so I agree strongly with regarding this as a mixture problem -- could be anywhere from almost 25% to a much much smaller fraction.
A variant on a box plot than can work well is a quantile-box plot, whether the quantile plot and the box plot are side by side, as in the examples I am going to show, or superimposed. Although the term quantile-box plot has been used in other ways, I go back here, at least in spirit, to a proposal by Emanuel Parzen in 1979. Some references are given in What's the history of box plots, and how did the "box and whiskers" design evolve?
I am not clear why we are being shown a simulation rather than the data.
Without attempting a close fake of the graph in the question, I knocked up a mixture that is a discrete uniform on integers 1 to 4 and random draws from a normal distribution with mean 50 and SD 10. The graphs below show situations (left) where the high callers are 20% of the distribution and (right) where the high callers are about 1% of the distribution.
Regardless of precisely what graph is used, the reader should be told how many callers are in each large group. Also, a transformed scale (square root of number of calls or logarithm of number of calls (plus 1 if zeros are present)) might help and not hinder visualization. Indeed, such a transformed scale would make it easier to see "inside the box" and note the detail of how many occurrences there were of 0, 1, 2, 3, 4, ... in the data.
Detail: Once it's been decided to show all the data the oversold rule "whiskers extend to the outermost points within 1.5 IQR of the nearer quartile" can be ignored. Here the whiskers extend to the extremes and are redundant given the quantile plot, but do no harm.
Detail: I don't use Python (hope to learn some one wet week with nothing else to do) so don't suggest code. Something like this graph shouldn't be difficult in any decent software.