OP,
He has said that he does not want to remarry. He is not keeping you a secret. If he does not plan to introduce you to his children, then he is right about your relationship being none of the ex wife's business. There would only be a need to know for her to know about you if you were in someway involved with her children.
Do you feel the need to inform your ex about this relationship? Do you feel the need to know whether your ex is dating or in a relationship?
There is a difference between privacy and secrecy. Obviously, he is not keeping you a secret if his friends and extended family know about you. As he has no intention of marrying or blending, he may feel tat that there is no need for his children or ex to know about you. That is simply keeping aspects of his life private from people who have no real need to know.
He may think of you as a romantic partner or team in a monogamous relationship but not in the sense of a life tying committed relationship.
You relationship may evolve into something different, but if you cannot accept this as all there may ever be then you are signing up for some up the road hurt and disappointment.
I would assume that your ex was very open and public about your relationship, and yet none of that openness led you to a good place.
If you are in a good place now, why try to force it into some preconceived notion of what it should be.? You aren't unhappy because you have not met his children. His children do not know about you, and assumedly they are happy in their uninformed state.
What would be gained by his introducing them to you?.
There is. no guarantee that you relationship status will change if they meet you, and a reasonably good chance that his relationship with them may be negatively impacted by introducing someone that he has no intention of living with or marrying. What would the children gain from this introduction at this early stage?
If you are not finding enough happiness in your current status to sustain your relationship, then maybe it is time to look for a man whose intentions better align with yours. Keep in mind that all change is not necessarily productive change and sometimes it is just activity that causes you to lose the positive things that you had .
Do whatever you feel that you need to do to maintain a life that brings you some joy and happiness.