Even if we believe that Shirley Williams was actually told by Blair that this happened, that still wouldn't mean Bush said it. Not least because Blair may have intended it as a joke, just as Williams may have.
This is still useful if our interest is "When was this said?" because we know Shirley Williams talks about it, so e.g. if some future historian is wondering about a claim Trump said this to Johnson, they've got this record that well, this particular "American Presidents are idiots" quote certainly pre-dates that.
There a lot of interesting turns of phrase where that is primarily what we care about - did Franklin say that stuff about hanging together or hanging separately? Perhaps not. This play on words does however seem to originate in that period, you won't find it much earlier than the American Revolution, while by the 19th century it's commonplace.