I love American English,can you please explain my motivation behind this blog?

“-/, or so he/she/they would have you believe.” implies that the precedent may be presented as a falsehood, lie, or misrepresentation.

In your example, it implies that Pytheas may be bearing false witness and was in fact an eyewitness to the event(s).

What makes Monsanto a “nice” corporation that makes safe products, or so they would have you believe,” implies Monsanto is (or may be) in fact, not a lovely corporation AND/OR does not make safe products? This could indicate direct mirepresentation by Monsanto, or by third parties (‘they’) such as new outlets or attorneys.

May also look a bit harder than you think.