You paint your ex as a 'gullible' and hapless innocent who has been taken in by the cunning of this women and her magic machine.
But in fact we are talking about a middle-aged man who is reckless enough to put the responsibility for the outcome of his entire future, and something he categorically did not want, in the hands of a 'girl' (how old is she?) that 'he barely knew'.
Are you aware of systems such as Persona and Clear Blue Fertlity Monitor? Both of these are machines that predict ovulation and indicate a women's most fertile times of the month. Although not (ostensibly) designed for the purpose, they are very commonly used as a form of 'contraception' by people who choose not to use other forms such as pills and barrier methods.
So, perhaps this 'girl' wasn't pulling the 'oldest tick in the book', but was telling the truth.
What I find breathtaking is that a grown man's interest would not have been stimulated to hear of such a marvellous invention...didn't he even ask about it, ffs?
Reminds me of Stewart Lee's despair of those people who, were you to leave them locked in an empty room, for 60 years, with nothing in the room apart from a tea-cosy, would be so lacking in curiosity that, through the course of the entire 60 years, it would not occur to them to put the tea-cosy on their head to find out if it would make a servicable hat.