Earlier quoted context omitted.
He can't make the evidence of absence claim, but he can absolutely make the absence of evidence claim.
Perhaps in an “everyday language” way, but not in the technical, statistical sense. In an underpowered statistical study, a claim that two experimental conditions did not differ are not persuasive.
The claim is not "two experimental conditions did not differ". The claim is "The data do not show evidence that the experimental conditions did differ".