This is the main quote from the Willmore interview you linked, that seems most relevant:
"I can only say that Mr. Musk, what he says, is absolutely factual. We have no information on that, though, whatsoever; what was offered, what was not offered; who it was offered to, how that process went. That's information that we simply don't have. So I believe him. I don't know all those details, and I don't think any of us really can give you the answer that maybe that you would be hoping for."
That has to be a nominee for doublespeak quote of the year -- and that's not a low bar this year.
I think the most likely scenario is that a message made its way up the grapevine to him along the lines of 'We're working with Elon to look at getting you guys brought back home on a Dragon.', nothing more, nothing less. NASA awarded a ~$270k contract to SpaceX on July 14th called "Special Study for Emergency Response." NASA claimed that study had nothing to do with Starliner, but that was probably a typical administrative lie of the sort Eric Berger has regularly pointed out (until this article...):
"NASA said this study was not directly related to Starliner's problems, but two sources told Ars it really was. Although the study entailed work on flying more than four crew members home on Crew Dragon—a scenario related to Frank Rubio and the Soyuz MS-22 leaks—it also allowed SpaceX to study flying Dragon home with six passengers, a regular crew complement in addition to Wilmore and Williams."
Nobody knows the details of exactly what was offered, or why it was rejected, besides Elon and whoever he was talking to in the previous administration. But at this point I think one cannot reasonable argue that no offer was made. And we know because of what happened that it was rejected. So the only question remaining is why.
"I can only say that Mr. Musk, what he says, is absolutely factual. We have no information on that, though, whatsoever; what was offered, what was not offered; who it was offered to, how that process went. That's information that we simply don't have. So I believe him. I don't know all those details, and I don't think any of us really can give you the answer that maybe that you would be hoping for."
That has to be a nominee for doublespeak quote of the year -- and that's not a low bar this year.