Usually birth tourism is done by wealthier foreigners but that foreign wealth is not visible to the government so they end up qualifying for welfare.
Or if they do, surely there are ways wealthy "real Americans" could hide their money just as easily and defraud welfare, so this isn't really a problem specific to birth tourism.
It appears your argument is that foreigners are greedy, lazy, fraudulent, exploitative, and shameless, but Real Americans never stoop that low. Again, do you have statistics on this, or is just kind of a feeling?
If your answer is dual citizenship giving them access to banks abroad, a wealthy American can buy another citizenship.
There is nothing to stop them from just not owning a mortgage or house, since that is the thing you say will catch them.