The motto is a cut from Horace, not a lab protocol.
Horace, Epistles I.1.14, to Maecenas: "Nullius addictus iurare in verba magistri" — not bound to swear by the words of a master. The next line is wherever the storm drives him, he puts in as a guest. He is refusing a philosophical school. He is not writing a rule for repeating an experiment.
John Evelyn's diary for 17 September 1662 records the Society resolving its arms "and the words Nullius in verba." Notes and Records of the Royal Society printed that entry in 1938. The second charter of 22 April 1663 grants the arms. The Society's own site now glosses the motto as a determination "to withstand the domination of authority and to verify all statements by an appeal to facts determined by experiment." https://royalsociety.org/about-us/who-we-are/history/
Those are two sentences. Horace dropped the master. The later gloss added the experiment. The essay treats the gloss as what 1660 was.
If Horace is the machine, "Trust the Science" does not invert 1660. 1660 was "don't be an Aristotelian." You can refuse a school and still take a fellow's word about what happened in the room last Wednesday.
Clive Sutton (British Journal for the History of Science, 27, 1994) argued the motto has been misread as "nothing in words / just the facts," which is not what Horace wrote. Competing account: the inversion the essay wants needs the later gloss. The two models differ on the next artifact. Restore Horace: stop making people swear to a paradigm. Restore the gloss: make a stranger able to repeat the demo. Those are different papers. A history of which schools the Society refused to join would support the first. A history of which claims died when a stranger repeated them would support the second.