It is common practice among nations to give citizenship to the children of their citizens, even if they happen to be visiting another nation at the time of birth. Would you go as far as to argue that a child who is a citizen of another nation is not "subject to a foreign power"?
Not at all common practice (at least, not universal). Not that it would matter, but my daughter was born in Mexico. While both myself and my wife at the time were nationals not of Mexico, but of another country (which I will leave unnamed), she had to be naturalized as a citizen of that country, which she was, at the age of some 2 months, following our formal petition to the Head of State of that country. Had we chosen not to petition the foreign government on her behalf, the naturalization would not have happened. Her Mexican citizenship, on the other hand, was immediate by birth.
In any case, this is irrelevant, since the relevant law here the US common law. As has been made very clear by Emsworth, this law disregards any possible outside allegiance (which might or might not exist by the national law of some other country - in many countries it does not, in fact, exist) and views any person born in the US (with clearly defined exceptions) as owing allegiance to the US and US alone.
Even in case of US not all children of US citizens born abroad are automatically citizens by birth. If I remember correctly, the legal provision requires at least one US-citizen parent to have lived some number of years in the US (an American colleague of mine actually had to prove that, by digging out his college transcripts and old tax returns, I think). Or, may be, the key is that the parent is US citizen by birth - I don't really remember. If that is not satsified, the child might still be naturalized as an American, but is not American by birth. This legal provision is established by law and could be changed, so, in principle, there is nothing that prevents the Congress from requiring naturalization of all foreign-born children of US citizens.
From this point my post is really irrelevant to the original question - just some extra thoughts on the matter.
I don't see the practical point of changing the existing practice. Generally speaking, current US immigration practice actually punishes the non-citizen parents of US-citizen kids, irrespective of their immigration intent. The child cannot "sponsor" his/her parents for legal immigration to the US until s/he turns 21 (after which it takes a lengthy family reunification process to achieve), but existence of such a child is viewed as evidence of immigration intent, which severely prejudices issuance of non-immigrant visas. Furthermore, existence of such a child provides no protection whatsoever from deportation for his/her parents (the existing provision that deportation might be waived to avoid causing "undue hardship to a US citizen" has been interpreted in the sense that it is not an undue hardship for the child to either follow the parents to their home country or to be left without them in the US). If anything, having a US citizen child makes a foreigner's life in the US more, not less, difficult.
So, the only incentive a foreigner has for giving birth in the US is the future welfare of the child. Given that the child is likely to be growing up culturally American (despite some claims to the contrary, most serious studies show that second-generation Americans today are as likely or more likely to be assimilated into the US culture as their predecessors 100 years ago), what the current practice avoids is creating a subclass of US-born and bred illegal non-citizens (they can hardly be called immigrants, since they never migrated). In fact, in many cases they might be impossible to deport, since, unless they voluntarily apply for some foreign citizenship, if they are not American, they would be stateless.
As an interesting (though irrelevant) contrast, even though at the time of birth of my daughter I had only been a non-immigrant foreign worker in Mexico (an equivalent of H1-B non-immigrant visa in the US) for about 2 years, once she was born I could immediately apply for the citizenship and it took me only about a year and a half to be formally naturalized as a Mexican (without ever having been a legal non-citizen immigrant in the country). There are countries with more relaxed laws!
Of course, one can take the opposite extreme. In Switzerland even a third-generation Swiss-born individual has to apply for naturalization after at least some 20 years of residence and have his/her name put on the popular ballot in his/her canton. They actually vote for every single naturalization (and much of the time they reject)! And in places like Korea and Japan naturalization is all but impossible (there's been some relaxation in Japan, but very minor), even after many generations in the country. And in Israel, not only is a child of legal foreign workers an illegal immigrant at birth, but his/her existence is grounds for revoking the parents' work visa, instantly turning them into illegal immigrants!
Just keep in mind, that in Japan, for instance, the result of their immigration practice has been a large number of North Korean-citizen residents, who go to North-Korea financed schools and express wholehearted allegiance to the Dear Leader. These people (Japanese-resident in 3-4 generations - since before WWII) would have long been completely assimilated, but for the rejection by the country of their birth.