I always find it amusing that the US insisting on something clearly political isn't a bad thing when others doing the same thing is. An organisation bowing to pressure for the US is fine, bowing to pressure from anyone else is the sign of a flawed, corrupt organisation. Meanwhile members of Trump's admin are bringing in wheelbarrows of cash to Florida to make sure that the fricking WWE is labelled as 'essential' and not subject to lockdown; which about sums up how things work and the expectations of Trump's America. You can also pretty much guarantee that the people complaining about the WHO now would be incandescent with rage if a similar approach to that they wanted taken with China were taken with the US. Indeed, we can already see that, China doesn't want it called the Chinese or Wuhan virus, the US does, China doesn't want Taiwan mentioned, the US does, and toys will come out of cot when either doesn't get their way.
WHO ain't going to call it Wuhan Flu because you call it SARS-CoV2 or Covid-19 or even generic old coronavirus for a reason- it's a coronavirus, not a flu. Flu is a, uh, orthomyxovirus, generically influenza/ flu (for good reason) and while they do share similarities with coronaviruses like ssRNA they are taxonomically different. Ironically the most prominent and deadly example of actual influenza was Spanish (sic) Flu which very likely originated in... the US and definitely didn't originate in Spain.
The US had relatively speaking plenty of time to prepare for covid19, the failure to do so effectively is largely on Trump downplaying everything (with a hearty dollop left over for D and other orange man bad types grandstanding on issues where Trump did do the right thing, like banning travel). China's behaviour has- as pretty much always- been awful with a policy of deliberate lies throughout but the WHO cannot simply invade China to get the true picture. They have also consistently been misquoted by Trumpians trying to run interference. Eg they did not say there was no human to human transmission, they said there was no evidence for it at that time (because, basically, China was lying about it- but they also had no particular reason to doubt them given that other recent coronaviruses with animal origin like SARS or MERS do have very low infectivity). The WHO issued a directive saying there was definite human to human transmission in early January iirc, about two months before Trump started taking covid19 seriously.
On a broader front and the broader issue, the US cannot complain about China's growing influence while simultaneously withdrawing from everything and anything half way multilateral they feel like because it isn't a tool for the US, Obama derangement syndrome or simple hubris. That just leads to handing China influence by default giftwrapped on a silver platter. One of the reasons there's so little international support for 'Wuhan Virus' is that pretty much everyone international literally literally hates Pompeo and Trump, and the US has already burnt through almost all of the goodwill that could have got support from Europe and others. No one wants to allow Trump the cover of blaming China, and given the low regard China is held in that really says something about how the current US admin is viewed.