So, to answer your first question, the CDC is not just an organization that works in the U.S: The CDC is actually a U.S. Government organization. The WHO, on the other hand, is not.
As to your second question, the WHO seems more dependent on the U.S. and the CDC rather than the other way around (emphasis mine):
CDC is a key partner of the Global Outbreak and Response Network (GOARN), which enables WHO to respond to acute public health events through the deployment of staff and resources. CDC has deployed 674 times into 53 operations and continues to provide support to DRC and surrounding countries amid the ongoing threat of Ebola.
As a founding member of GOARN, the CDC is part of its steering committee and leads on various aspects of the GOARN 2.0 strategy. Other US-based GOARN partners include ProMED, Harvard Humanitarian Initiative, Resolve to Save Lives, Global Virus Network, MWater, Chan Zuckerberg Biohub, Johns Hopkins University, UNICEF, WHO Regional Office for the Americas, US Army Medical Research Institute of Infectious Diseases, and TEPHINET.
US funding has stopped the spread of disease and helped people stay healthy in thousands of humanitarian crisis and disease outbreaks...
Of course, don't take this to mean that the WHO's relationship with the CDC and the US is entirely one-sided. The CDC does appear to get help from WHO (again, emphasis mine):
In close partnership with WHO, the US Centers for Disease Control and Prevention (CDC) facilitates regional training events on foundational principles and concepts in public health emergency management. In 2018, a total of 79 participants representing 23 countries attended regional training events in the WHO AFRO region.
And, according to the CDC, the WHO does help with influenza monitoring:
The World Health Organization’s Global Influenza Surveillance and Response System (GISRS)external icon was established in 1952 to monitor the frequent changes in influenza viruses with the aim of reducing influenza disease impact through the use of vaccines containing currently circulating strains.
Though, reading further, it's unclear who helps who more: the CDC or the WHO (try saying that out loud 5 times fast).
Anyway, the point is, I see no reason why the CDC couldn't pick up the slack if WHO suddenly wasn't involved, though keep in mind that the CDC and the WHO still seem to have a strong relationship in spite of President Trump's actions to cut funding:
“CDC and WHO has had a long history of working together in multiple outbreaks throughout the world, as we continue to do in this one,” CDC Director Robert Redfield said Wednesday on “Good Morning America.” “And so, we’ve had a very productive public health relationship. We continue to have that.”
So that may never happen anyway.