Interesting how you've taken the very essence of what "virtue signalling" is i.e. the lazy fact-free thinking of people who try to put someone down (and appear morally superior in the process), and somehow convoluted that to suggest the lazy, fact-free part is actually a fault of those who call out the actual virtue signallers themselves. IMO such a convolution isn't a fair interpretation at all.
I think you overlook the ability of the "average punter" to understand the nuances of virtue signalling and recognise it without needing to have every instance of it explained or demonstrated to them in detail
ad nauseam.
Like it or not, "virtue signalling" is a great descriptor which has, through necessity, become part of the moden lexicon. Most people can recognise a virtue signaller in an instant - picture someone (usually, but not exclusively, from the political left) shouting down (abusing?) anyone who doesn't share their views and calling them either islamophobic, racist, homophobic, or abusers of women (or maybe all of the above at the same time)
just because they don't share their views. Worse still, virtue signallers not only don't bother with reasoned facts themselves; they also wilfully dismiss the reasoned facts of others and their right to hold different opinions.
If pointing out the shortcomings of virtue signallers is a put down, then it's an unavoidable consequence of the behaviour. One thing is sure, the term 'virtue signalling" is here to stay simply because there is an increasing need for it!