I'm no fan of the Chinese government, and I think the WHO has taken some pretty lousy political decisions in the past few years (it screwed up on Ebola before this). I also agree that the WHO has deferred too much to China on this. But much of this popular story you refer to seems to me to be an effort by governments to shift blame and excuse their own negligence.
China contributes a relatively small portion of its funds. You know who contributes more? The US. And the UK. And Germany. And Japan. And the EU. And the Gates Foundation. Even goddamned Rotary International pays more than China.
Of course, maybe money is not the main factor. After all, the director-general was elected with support from China. So who runs the WHO? It has 194 member states and an executive board of 34 members that is chaired by a leader that rotates geographically. The chair probably doesn't have much power, but FWIW, it was led by Japan till a week ago - hardly a shill for China. Where's the US in all this? Oh, it's not bothered to nominate anyone to its executive board seat, which has been left vacant for years. This is consistent with the US government's choice to ignore or weaken pretty much every international organisation. If the US chooses to ignore these organisations, is it any wonder that other entities increase in influence, if only by virtue of being at the table?
The WHO made statements in January and February that were stupid and wrong, even given only the information available publicly. But you can call it a strategy: they were probably trying to coax China into giving their experts access to Wuhan to study what was going on. To some extent it worked, China did let them in. It was not worth the damage caused by the misinformation. But the WHO has no power to force any government to do anything, so they probably thought mollifying the Chinese leadership was the best option they had. Especially when the only powerful country that could back them in this fight, the US, was in absentia and busy praising the Chinese leadership for their efforts.
Most importantly, however, plenty of countries did get the obvious message back in January and February and took action. Vietnam has had 0 deaths and 300 cases. Mongolia has had 0 deaths and 200 cases. These are relatively weak, poor countries that share a border with China. They simply jumped into action immediately, based largely on information that you could read in any international newspaper at the time. Even Kerala had a plan in place in January.
So whatever influence China may have at the WHO, most of the lives lost have been because of the incompetence of authorities in the US and Europe. This was not inevitable. There was more than enough information available in January and February to take action, and for reasons that I do not understand, most governments and health authorities twiddled their thumbs.
The WHO is a convenient scapegoat for their failures to save the lives of hundreds of thousands (likely over a million before this is over) and the livelihoods of hundreds of millions.