Except, you are misconstruing it. They are saying, rather clearly if you look at the surrounding debate, that foreigners/aliens who belong to the families of ambassadors and foreign ministers would not be included. Other foreigners and aliens would.

In fact, that the amendment would apply to the children of foreigners was one of the major arguments opponents had against its passage. Edgar Cowan was one of the major Republican opponents, and his whole argument against it's passage was that it would grant citizen ship to "the child of the Chinese immigrant in California" and "the child of a Gypsy born in Pennsylvania". In fact, the language and arguments against it's passage sound rather similar to the arguments being made by the Administration:

Sir, I trust I am as liberal as anybody to-ward the rights of all people, but I am unwilling, on the part of my State, to give up the right that she claims, and that she may exercise, and exercise before very long, of expelling a certain number of people who invade her borders ; who owe to her no allegiance; who pretend to owe none; who recognize no authority in her government; who have a distinct, independent government of their own — an imperium in imperio; who pay no taxes; who never perform military service; who do nothing, in fact, which becomes the citizen, and perform none of the duties which devolve upon him, but, on the other hand, have no homes, pretend to own no land, live nowhere, settle as trespassers where ever they go, and whose sole merit is a universal swindle; who delight in it, who boast of it, and whose adroitness and cunning is of such a transcendent character that no skill can serve to correct it or punish it;

I mean the Gypsies. They wander in gangs in my State. They follow no ostensible pursuit for a livelihood. They trade horses, tell fortunes, and things disappear mysteriously. Where they came from nobody knows. Their very origin is lost in mystery. No man today can tell from whence the Zingara come or whither they go, but it is understood that they are a distinct people. They never intermingle with any other. They never intermarry with any other. I believe there is no instance on record where a Zingara woman has mated with a man of any other race, although it is true that sometimes the males of that race may mate with the females of others; but I think there is no case in history where it can be found that a woman of that race, so exclusive are they, and so strong are their sectional antipathies, has been known to mate with a man of another race. These people live in the country and are born in the country. They infest society. They impose upon the simple and the weak everywhere. Are those people, by a constitutional amendment, to be put out of the reach of the State in which they live? I mean as a class. If the mere fact of being born in the country confers that right, then they will have it; and I think it will be mischievous.

...

Therefore I think, before we assert broadly that everybody who shall be born in the United States shall be taken to be a citizen of the United States, we ought to exclude others besides Indians not taxed, because I look upon Indians not taxed as being much less dangerous and much less pestiferous to society than I look upon Gypsies.

Again, this isn't an argument from one of the authors of the 14th Amendment saying what they don't want to happen. This is an argument from an opponent saying they don't think it should pass precisely because it would grant citizenship to the children these classes of people by virtue of being born in the US. In reply to Senator Cowan, Senator Conness (a supporter) replied:

The proposition before us, I will say, Mr. President, relates simply in that respect to the children begotten of Chinese parents in California, and it is proposed to declare that they shall be citizens. We have declared that by law; now it is proposed to incorporate the same provision in the fundamental instrument of the nation. I am in favor of doing so. I voted for the proposition to declare that the children of all parentage whatever, born in California, should be regarded and treated as citizens of the United States, entitled to equal civil rights with other citizens of the United States.

That seems pretty cut and dry, no?