> Taiwan is not part of WHO due to China's efforts to keep them out.

Taiwan isn't part of the WHO because the WHO is part of the UN and broadly speaking, member states of the UN -- the United States included -- don't officially recognize Taiwan.

As an aside, the US relationship with Taiwan is entirely unofficial. The US political stance follows the One China policy and recognizes Beijing as the capital of a China inclusive of both what is colloquially the PRC and the ROC.

For what it's worth, it's likely due to the close (if challenging) relationship the PRC and ROC enjoy that Taiwan was able to get a head start on containment.