In the phrase “toxic call-out culture,” I’m interested in the word “toxic.” The last time it was part of a widely used set phrase was back in the good old days of the Global Financial Crisis, when we heard a lot about “toxic assets.” The theory was that if those individual assets could be isolated and excised, then the banks would go from strength to strength. But the metaphor already gave the game away — it is in the nature of a “toxic” element to pollute everything else. And in reality, the toxic assets may have had higher toxin levels, but they were a symptom of a system that was toxic overall.
Similarly, the idea in “toxic call-out culture” diatribes is that if the left could get rid of this one element, they would be successful. But again, the very metaphor implies that the entire left is polluted already. And even if we concede that “call-out culture” gets out of hand or becomes an area of disproportionate focus, such events are only particularly vocal instantiations of core convictions of the left — above all, a preferential option for those historically oppressed and excluded, as well as the recognition that those from historically more privileged groups need to be held accountable if they want to participate in the struggle for liberation.
On a local tactical level, some particular instance of “call-out culture” could certainly be judged counter-productive or ill-advised, but on the level of global strategy, getting rid of the impulse behind “call-out culture” would mean giving up on being the left at all. If call-out culture is “toxic,” then we can only conclude that the left as such is toxic, as it surely is for those commentators who make use of the trope.
An und für sich 
The impulse behind call-out culture produced ridiculous spectacles like /r/socialism banning the creator of its own banner because she also creates this comic: http://politicalideologycatgirls.com.
I called it a ridiculous spectacle, you might call it ill-advised – we might also call it toxic, in which case it falls under the logic of your first paragraph.
Left wingers expect more from one another, and rightfully so. Call-out culture is entirely irrelevant to a loud-mouth conservative; it has the power it has because we want to hold *ourselves* to a higher standard. But that power can turn into a religious logic of accusation, confession, and penance. No one wants that, so avoiding it means that when a call-out appears necessary, we all ought to remember that we should expect good faith, and reject confession and penance.