I'm a US expat living in Canada, and a parent of Canadian children. They are eligible for US citizenship, but don't become US citizens unless I go through a procedure to register them as such.
That's not true; they are in fact citizens under US law. They may not have an SSN or a passport, but neither is a condition for citizenship.
Also, at least from a tax perspective, if you plan on remaining a US citizen, you might benefit from getting SSNs for your kids so you can claim the child tax credit and stimulus payments for 2021.
And if they don't want to be citizens, renouncing citizenship in the 6 months immediately after they turn 18 can avoid tax complications.
