Your quote said that there could be problems in establishing citizenship. So apparently, the citizenship is only established (in the meaning of confirmed?!) when the parents or child actively seek this citizenship (and meet the requirements); acvording to your quote only then it is truly acquired and the rights that come with it can be exercised.
Not exactly. According to the quote, citizenship is truly acquired at birth but the rights can only be exercised after documentation is actively sought and granted.
Here are the U.S. Department of State quotes (and links to the whole texts) again, as I posted them pages earlier in the thread.
Birth of U.S. Citizens Abroad
Parents of a child born abroad to a U.S. citizen or citizens should apply for a CRBA [Consular Report of Birth Abroad of a Citizen of the United States of America] and/or a U.S. passport for the child as soon as possible. Failure to quickly document a child who meets the requirements for acquiring U.S. citizenship at birth may cause problems for the parents and the child when attempting to establish the child’s U.S. citizenship and eligibility for the rights and benefits of U.S. citizenship, including entry into the United States. By law, U.S. citizens, including dual nationals, must use a U.S. passport to enter and leave the United States.
Acquisition of U.S. Citizenship by a Child Born Abroad
Birth Abroad in Wedlock to a U.S. Citizen and an Alien
A person born abroad in wedlock to a U.S. citizen and an alien acquires U.S. citizenship at birth if the U.S. citizen parent has been physically present in the United States or one of its outlying possessions prior to the person’s birth for the period required by the statute in effect when the person was born (INA 301(g), formerly INA 301(a)(7).) For birth on or after November 14, 1986, the U.S. citizen parent must have been physically present in the United States or one of its outlying possessions for five years prior to the person’s birth, at least two of which were after the age of fourteen. [...]
Both quotes state clearly that the citizenship is acquired "at birth".
In the first quote, "establish" is used with the meaning of "prove". You may be familiar with the legal term "establish beyond a reasonable doubt".
Potential problems with entry into the U.S. are also discussed in the first quote. These problems are possible only because undocumented citizens are nonetheless considered citizens in law. If citizenship were only acquired after active documentation, the undocumented child would be considered a foreign citizen only, and the rule prohibiting U.S. citizens from entering United States territory on a foreign passport would pose no problems for them.
Since people on this forum tend to post partial information
This is the whole text from the US Department of State, Bureau of Consular Affairs
Birth of U.S. Citizens Abroad
A child born abroad to a U.S. citizen parent or parents may acquire U.S. citizenship at birth if certain statutory requirements are met. The child’s parents should contact the nearest U.S. embassy or consulate to apply for a Consular Report of Birth Abroad of a Citizen of the United States of America (CRBA) to document that the child is a U.S. citizen. If the U.S. embassy or consulate determines that the child acquired U.S. citizenship at birth, a consular officer will approve the CRBA application and the Department of State will issue a CRBA, also called a Form FS-240, in the child’s name.
According to U.S. law, a CRBA is proof of U.S. citizenship and may be used to obtain a U.S. passport and register for school, among other purposes.
The child’s parents may choose to apply for a U.S. passport for the child at the same time that they apply for a CRBA. Parents may also choose to apply only for a U.S. passport for the child. Like a CRBA, a full validity U.S. passport is proof of U.S. citizenship.
Parents of a child born abroad to a U.S. citizen or citizens should apply for a CRBA and/or a U.S. passport for the child as soon as possible. Failure to quickly document a child who meets the requirements for acquiring U.S. citizenship at birth may cause problems for the parents and the child when attempting to establish the child’s U.S. citizenship and eligibility for the rights and benefits of U.S. citizenship, including entry into the United States. By law, U.S. citizens, including dual nationals, must use a U.S. passport to enter and leave the United States.
This text is the same as the text of the article "Birth of U.S. Citizens Abroad" linked to in my earlier post (message #1009, reposted above). Thank you for posting it again. In the first paragraph, the text states that citizenship may be acquired
at birth (not upon documentation) if statutory requirements are met, with the implication that it may not be acquired at birth if statutory requirements are not met. Further, it describes the CRBA as proof (not acquisition) of U.S. citizenship.