What's really disgusting is people taking public transportation and expecting to be treated as if they were on a personal jet. Also, people who think that just because they have kids their interests must be seen before everybody else's.
However unpleasant the result is to the OP, EK did precisely what they were supposed to do - when they saw a possible health threat (motion sickness is not the only thing that can cause vomiting), they removed the pax from the airplane so that he could be seen by a doctor.
Should have EK held the whole plane until the pax was examined? No. There were a lot of other people on the plane for many of whom departing on time was important.
Should have EK paid for the hotel room? No. The OP trip interruption was not EK's fault. The OP should have had (and probably have had) trip insurance that would possibly have covered the trip interruption costs.
Should have EK accommodated the OP on the next earliest flight? No. Again, the OP trip interruption was not EK's fault.
So, what's the problem here? The alleged impoliteness of EK's employees towards the OP? But it seems that impoliteness wasn't the point of contention - the OP seems to be upset because EK removed them from the plane (which was the right call), and didn't pay for their hotel (which EK wasn't obligated to do), and didn't buy them a ticket on QF (which EK wasn't obligated to do).
So, please stop acting as if EK didn't give you something they owed you. What would happen if EK let you fly and Max had complications mid-flight? Would you not be posting here that bad EK let your family fly instead of doing the right thing and sending Max to see a doctor?