#1 You already know the answer of what you should do...when you said "I know what my children would prefer". Your kids did not ask to be born. That was a choice you and your ex-husband made. So as someone responsible for their lives, would you not want to ensure that they have the best life they could possibly have?
#2 As a woman it is a HUGE flag to hear that your boyfriend thinks that having events and doing things together with the father of your children is now inappropriate. Your ex will always be the father of your children and I'm sure your children will always desire their father's presence. I would take a guess and say that a lot of children would be happy seeing their parents successfully and peacefully co-parenting and involved with major events such as birthdays. I know I was happy when I had both parents present even though they were divorced.
It leaves me with concerns about his maturity when you said, "My boyfriend asserts that we are a broken family and should handle everything separately" and "If he wasn't good enough to be married to, why would he be good enough to share special days with?". It sounds as if he isn't empathetic towards your children's interest....
#3 My mother and father never exposed me to their dating life, if they dated (again I wouldn't know because I wasn't exposed). This granted them time and energy to spend with me and my needs. Personally I was grateful for this.