Is there someone who had a happy ending with an avoidant (while struggling at the start of a relationship)?

Struggling here. I (37F) have been dating someone (39M) who has admitted to have some fear of attachment (he thinks it is because of a very bad break-up with his ex, whom he saw crashing emotionally). On top of it, he is quite insecure and has kind of a fragile ego. But at the same time, he is a wonderful listener, very sensitive and a beautiful person. He feels guilty quite quickly towards people.

The first 2-3 months, the dates were very consistent, once a week and very nice, he loved our connection, he said (we never escalated to sex).Then all of a sudden, he said he felt physically there was a detail missing, and although he hoped he would forget about it thanks to our connection, it continued to play in his mind. But he wanted to keep in touch. We just started to see each other as friends then, while he always initiated to see each other (if I asked to see each other, it never worked out). But things got warm again a bit and I asked for clarity on the phone because I wanted to go on with my life and I wanted closure, and he told me "he doesn't think we would end up together". I said ok. But 2 hours later he sends me a text message to meet again. But before the meeting could happen, he asks me to reschedule it. I was fed up and said I was not free on the suggested date, and that he could let me know when it would actually suit him one day (which for me is a cold approach compared to normally, because I have always suggested an alternative in the past). I guess he feels I am done with it and he did not reply anymore.

My mind says to let him go because it is complex already and because he clearly told me he didn't see it going anywhere. On the other hand, everytime he says something like that, a bit later he always reaches out to meet each other again, so his actions contradict his words. He also admitted he has a tendency to leave things in a "grey zone".

But my heart is crazy after him. Is there hope? Should I reach out again and fight for him and show patience, or should I believe his words? Has anyone had a difficult start with an avoidant person, but today has the proof that it can work out?