[T]he World Health Organization offered [...] the testing kits that they have available and to give it to us now. We refused them. We did not want to buy them. We did not want to get them from them. We wanted to make sure we had our own. I think [Trump] said something like we have the best scientists in America, or something to that effect.
I'm not sure what would be a truly convincing level of evidence for this claim. Also there's the semantic distinction between "not asking" vs "being offered and refusing". But if we gloss over those issues, the topic that Biden has mentioned has been covered in the press before, e.g. by NYT on March 2
Soon after the virus surfaced in China, the C.D.C. got to work on its own test. “Generally, C.D.C. provides these tests for the world,” said Dr. [Thomas] Frieden[-- a former director of the C.D.C.].
But German researchers were devising their own test, which was quickly adopted by the World Health Organization for distribution around the world.
After the C.D.C.’s version turned out to be flawed, the agency continued to pursue it, despite the fact that another diagnostic test was already in wide use.
With F.D.A. approval, the agency could simply have distributed the test used by the W.H.O. instead of creating a new one from scratch, Dr. Mina [an epidemiologist at Harvard University] said. The government could do so even now.
“It’s just a very American approach to say, ‘We’re the U.S., the major U.S. public health lab, and we’re going to not follow the leader,’” Dr. Mina said.
So yeah, the US (or more precisely the CDC) apparently had a NIH syndrome (non-invented-here) in this matter for a certain amount of time, with respect to the foreign-developed tests endorsed by the WHO.
(Am I merely repeating the claim [in an earlier version] rather than proving it though? You decide.)
The New Yorker has recently published (on March 16, more precisely) a more detailed account on the timeline of the competing tests and the complications of US (FDA) regulations with respect to their actual use and even their advertisement (in the US). These EUA regulations affected tests developed in the US too (outside of the CDC.)
As for the delay in scaling up covid-19 testing capacity during those crucial weeks in February, [Keith] Jerome [--the director of University of Washington Virology Lab--] told me that the underlying problem had far less to do with the faulty tests produced by the C.D.C. than it did with a system that could not contemplate, let alone manage, the possibility that the C.D.C. might end up producing faulty tests. The F.D.A.’s exclusive authorization to the C.D.C. to conduct covid-19 tests ended up creating “what you’d think of as an agriculture monoculture. If something went wrong, it was going to shut everything down, and that’s what happened.”
As LShaver points out in a comment below, and as I suspected (in my opening sentence) the "WHO offered but US refused" isn't true:
The Biden campaign referred us to a Politico article that said the WHO shipped coronavirus tests to nearly 60 countries at the end of February, but the U.S. was not among them. That is technically correct, but it suggests that the United States would have been on the list under any circumstances. [...]
"No discussions occurred between WHO and CDC about WHO providing COVID-19 tests to the United States," said WHO spokeswoman Margaret Harris. "This is consistent with experience since the United States does not ordinarily rely on WHO for reagents or diagnostic tests because of sufficient domestic capacity."
So "US didn't ask" is what really happened, as I suspected.