N.W.
Let me tell you my experience, as a child whose father left her...
My dad left my mom, sister, and I when I was two - to be with his new girlfriend. Over the next year his parents would take us to their home occasionally to visit and fill our heads with the same junk you describe - telling us my dad wanted to see us and that my mom wouldn't let him, how great he was, etc. And, on the off chance my dad would pay us a visit at his parents house, or perhaps even take us for an evening, he too would make us so want to be a part of his life.
Well, after my sister and I coming home in tears, crying for several days after that we missed our dad, she finally called an end to it.
My dad ultimately gave us up for adoption to the new man in my mom's life two years later and from that moment on he and his family never looked back. And, before he signed us over, my mom told his parents that if they couldn't see us without talking about our dad that they would no longer be able to see us... so they chose then not to see us.
All these years later - I was two when it started, my sister was four - I am now 36 and I totally respect my mom and her decisions to get that man and his family out of our lives. She did what she had to do so that our lives could survive on the normal everyday ways. We, and she, did not need that drama.
I was about 13 when I contimplated trying to find my dad and contacting him... but that quickly passed when I realized I'd be looking for someone who didn't care or love me and that all I would be would be heartbroken.
So, you do what your mom instincts tell you to do. You can't go wrong doing what you think is right for your child.