Would my children have to pay FairTax when making purchases?
The FairTax is just a version of a GST.
SoftTalker then noted that we tax resident aliens but do not allow them to vote, presumably seeing some similarity in terms its impact on the (implicit) social contract.
What do you see as the difference?
Yes. That is a contradiction. Categorically, it's not fair to expect someone to pay for things without letting them have some degree of decision making in how the money is spent. Otherwise it's simply robbery.
1. not taxing people who live and work here but are not citizens?
OR
2. allowing people who are not citizens to vote?
Unless we do one of these, I'm fairly sure the situation you describe as "messed up" will continue to exist.