Some background check and analysis of COVAX:
The composition of COVAX is WHO, GAVI, CEPI and U.N. Chinldren's found.
WHO as a professional medical agency is like the regulators in countries. BUT with members from member states.
GAVI's primary financiers are in high to low order, UK, Bill Gates (USA), USA and Norway.
CEPI's funding order: Norway, Germany, Japan, Bill Gates (USA), European commission, UK, Canada etc.
Now, interesting look, isn't it? All the power (money) are mostly in the hands of overt anti-China hands or some minor covert anti-China hands. It is not a organization that China should get involved at all. Don't jump into the trap for others to beat you. It is just another G7. We know what happened to Russia when it was in the G8.
WHO as an "hired" bureaucrat can NOT act neutrally and professionally even if its members wanted to (I believe most of them do). Just look at what happened to Dr. Faucci.
So my thought is that, China's pledge to work with COVAX (providing vaccine) should be (or indeed being) lukewarm and perfunctory. That is, make a position of cooperation as a responsible country, but don't provide satisfactory data to get China's vaccine into the program (knowing the "partners" are going to bite you at every moment).
There are better ways to really contribute the world's population who deserve and appreciate China's effort, do it through bilateral channels. China can and is willing to work through western built systems, but if not treated fairly, China should walk its own way, build alternative world orders, just like Asian Infrastructure Investment Bank and New Development Bank against Asia Development Bank, World Bank and IMFs.
My final conclusion is that "I wish China's relation with COVAX to fade into obscurity".