For **** sake. If they had been allowed to travel and the vomiting had turned out to be caused by something serious, the airline would be vilified for not exercising an abundance of caution. Of course they're going to offload someone vomiting - they're not doctors, expected nor trained to make the judgement on medical cases! No, kids don't normally "throw up, especially when travelling" as a rule. Vomiting can indicate anything from something completely benign to something extremely serious. Flight safety is all about risk management, take your pukey kid and look after them somewhere more suited to do so than a enclosed metal tube hurtling through the air.
It would seem the situation was handled clumsily afterwords, but that doesn't make the decision to offload wrong whatsoever.
Yes, it's a shame that their child's sickness ruined their holiday but guess what - they're nowhere near the first or the last family for that to be the case. Trying to pin it on someone or something else is just a sign of the times we live in.
As far as self-appointed "passenger advocates", clueless as usual.