could the US make a person a US citizen specifically to be able to
charge them with a crime and possibly extradite them.
This part of the question is based upon a false premise.
Extradition has little or nothing to do with citizenship.
If someone who is currently in the hypothetical country of Albia commits a crime under the laws of the hypothetical country Elbonia, and Elbonia requests that the person suspected of committing an Elbonian crime be extradited by Albia to Elbonia to face those criminal charges, Albia can and not infrequently will agree to extradite the suspect to Elbonia without regard to the suspect's citizenship.
Very few countries prohibit the extradition of their own citizens for crimes committed abroad which would be comparably serious crimes not subject to the death penalty in a foreign state. Likewise, one does not have to be a citizen of the state in which the suspect allegedly committed a crime in order to be extradited to that state to face criminal charges.
Generally speaking, Elbonia cannot deport Elbonian citizens in Elbonia for committing Elbonia crimes. But that doesn't prohibit Elbonia from extraditing an Elbonian citizen to Albia for committing an Albian crime.
Each country determines its own rules for determining who is a citizen of that country and who is not.
These rules are frequently inconsistent with each other. On one hand, that means that it is frequently possible to be simultaneously a citizen of more than one country. On the other hand, that means that there are circumstances under which a person can become "stateless" and have citizenship in no country.
Most of the time, your citizenship is not a matter of personal choice. The vast majority of the time, your citizenship is initially automatically established in one or more countries at birth, either by the place where you are born, the citizenship of your parents, or both. Neither you nor your parents have any say in determining your citizenship at the time of your birth in most cases.
Once your citizenship is established at birth, a new citizenship may established by naturalization, and an existing citizenship may be renounced. Naturalization opportunities are frequently heavily regulated and limited, and there are significant barriers to renouncing one's citizenship, in part, to avoid the problem of people becoming stateless.
I think I am asking - is it really entirely up to the country to
proclaim someone a citizen?
A country could decide under its own laws that you are have become its citizen without your input, and indeed, this actually isn't all that uncommon.
The U.S. retroactively made lots of people citizens of the United States when the 14th Amendment to the U.S. Constitution was adopted.
Many countries retroactively declare that people with ancestors who were citizens of that country are citizens of that country.
When can you renounce citizenship? Einstein famously renounced German
Each country can decide when and under what circumstances its citizens can renounce their citizenship for purposes of its own laws. Einstein's renunciation of German citizenship probably wasn't legally valid under the laws of the Third Reich in Germany.
But other countries can treat someone who renounces a foreign citizenship as no longer a citizen of that country even if the country whose citizenship is renounced doesn't allow its citizens to do so. Einstein's renunciation of citizenship was treated as valid under U.S. law without regard to the position that the Third Reich in Germany would have taken on the question.