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Dear LW,
Let me first acknowledge the difficulty, the emotional crush, of your situation. I hope you've read through the many thoughtful responses readers have posted here, and ignored Cary's misguided if gentle ramblings about men needing time, etc. before making these sorts of fundamental decisions that govern the course of a relationship. Yes, thinking and evaluating is wise. But your significant other is displaying a distinct indifference, and coldness, in response to what is a perfectly reasonable request two years into an adult relationship. You have asked to meet his daughters, to be more fully in his life. He simply does not want this.
I have quoted Will Shakespeare above on the nature of love -- love shows itself, love acts, love moves, love listens. Your significant other has a dubious record and may indeed have pain and shame related to the broken engagement as Cary suggests, but more likely, and especially at his age, he has a long-standing, deep streak of self-centeredness that poisons his relationships.
As you indicate, you are over 50. Frankly you deserve far more and far better. It is time to move on, firmly, absolutely.