I don’t think you should force friendship until you are ready to. You can co-parent perfectly well with minimal communication until you are ready to communicate more without it hurting a whole lot.
When my last long term relationship ended she wanted to be friends after. I was still really hurting and those random texts and phone calls and invitations to get coffee hurt me and reminded me of what I had lost on a regular basis. I went no contact for about half a year, when the mourning process was over and I had accepted that I was better off without her, it felt okay to be back in contact.
So I disagree with the other posters - protect your heart. Your kids will be okay. Your kids want a parent who is happy. They can’t have that if you’re forcing friendship contact with your ex that is hurting you and prolonging the grieving/healing process. Go minimal contact, take care of your heart, rediscover how to be happy as a single person, then when you’re ready you can talk to your ex more regularly, if you ever want to.
As a child of divorce, what was more important to me was that my parents were happy. Kids can tell when their parents are hurting. And my parents being friendly never felt that important. You can be friendly and nice to each other at pick up and drop off without having to text and have phone calls and get coffee outside of strictly child related interactions. Your kids won’t know if you and your ex are texting regularly. They will know if you are sad.
You are worthy of real, earnest love. You deserve someone who wholeheartedly loves and desires you. You can find that person as soon as you end this relationship and finish mourning the end of the relationship you had. Breakups are a form of death in your life, but sometimes you have to let things go to be free for the next good thing in your life. Don’t waste your time on lukewarm partners.
I’m 36. It worked with 5ish women before I met my wife (worked with her too) while dating in my 30s. The sillyness is in some ways the point. Don’t take it too seriously, and only ask if you know the answer. Humans give lots of social cues when they want to be intimate: eye contact, smiling, touch, scooting closer, flushed cheeks, incidental contact of the knees or hands. Read those cues correctly and it won’t really matter what words you use because you will already know they want to. And humor is attractive.
For whatever reason, probably societal conditioning, women tend to expect the man to initiate intimacy. It’s scary and nerve-wracking and a lot of pressure. But just throw this line out there. The worst thing that can happen is they say no thanks and you move on.
I’ve found that women, even and perhaps especially adult women, enjoy playfulness, fun, and jokes. The world is serious and difficult. Dating can be a huge pain in the ass. Finding a partner that makes you laugh is often one of the top things women are looking for in a partner.
When you’re sitting close to each other, literally say, “do you want to make out for a little while?” Only do this if you know what the answer will be. I like this because it’s not some super corny line, but it’s still kind of funny. Not everyone wants you to grab them and kiss them like in the movies. It’s important to get consent. This silly line has worked for me every time, because you should already know the answer. They’re usually just waiting for you to ask.