Your suggestion that there was a twenty or so-year gap with this one does not match this pattern.

I think that Petunia getting the beauty potion was a key event that caused fate to head in that direction. If she didn't she would be married to Dursley and Harry wouldn't get his rational edcuation.

It is, however, a prophecy about Harry, whether anyone knows it or not, and all other prophecies about Harry were pronounced not long before or after the key events that caused fate to head in that direction

That's probably not true. A lot of students have divination classes at hogwarts and make predictions on a constant basis.

I find it very unlikely that those devination classes all passed without any predictions that involved Harry.

Most future telling in the Harry Potter universe isn't about prophecies of the kind where Trelawney goes into a trance and speaks the words of the prophecy.

Centaurs don't do future telling that way. They do astrology which can take years to interpret.

And if her objective had been to persuade Petunia that she was telling the truth, she'd have taken the time to explain what she'd been told until it no longer sounded like a ridiculous excuse.

If I told you that a friend of my talked to his spirit guides and they told him, that I should write you an email, how long would it take for me to explain to you the fact in a way that doesn't sound ridiculous?

You also seem to somehow assume that people persue every goal that they have with maximum energy. That isn't the case.