Honestly, I’d say yes, no, and maybe. It truly depends on the situation.
Yes, because where they exists, two mature adults know that they should be and were friends first. Breaking up between them is mutual; no one dumped the other for bragging rights. It also should never be a nasty breakup. People should always try their best to sever matters on great terms while remaining truly good friends. So much bad things are happening these days that the last thing you want is to end up carrying around guilt for possessing regret, unable to say that you’re sorry all over a breakup that ended horribly for petty reasons. Even better if you’ve moved on and your new partner embraces your friendship with your ex, as long you they are included as well, being they’re a new part of you.
No, because the world isn’t perfect, therefore we must sometimes deal with those that are immature, uncompromising, unruly, bitter, overheated, temperament, and stubborn no matter what anyone says or tells them. Unfortunately, sometimes we fail to have those characteristics of the other person exposed or revealed to us before breakup time arrives at our doorstep. Then their true colors decides to go Picasso Pro, showing so they really are deep down. At that point is when the irate one’s decisions should be respected to the fullest and simply walk away without, disdain, hate, or malice on the heart and mind. Let them be. It’s better to let them alone and angry without speaking than angry and causing someone injury or worst. Try to be aware and remain cautious of those you deal with.
And maybe, because it depends heavily on your/their new partner you’ve/they’ve decided to move on with. It also depends on if that ex should be considered to be trusted as a friend afterwards based on their actions that led to the breakup, if applicable. It’s easy to say that everyone should get along as adults and be mature about having exes remain as friends. Which is true overall, but what makes us unique individually is our personality, characteristics, morals, likes, dislikes, beliefs, principles, and boundaries. Therefore, we should respect and embrace one another’s uniqueness. That doesn’t mean to just put up with anything and fall for the obvious of everything else. Respectfully, walk away if you are unable to submit and commit to another one’s morals and principles in particular.
Hint: This is why it would behoove people to date first. That way conversations can be had on most important matters. Give the dating title a deadline. Then, by that time, two will have figured out if they’ll willingly compromise mutually coming to agreements on relevant aspects and make things work. Or not.
If you know your new partner wouldn’t or doesn’t approve you and your ex having that buddy system, then respect them to the fullest. It’s not that your new partner is necessarily insecure or jealous, but rather in the mindset of protecting their precious jewel from being lost to someone that already knows them well. Because it happens. Or, still respect them by respectfully walking away if it’s something you know you wouldn’t own the capability of doing. Under these circumstances, things only become difficult because people make them difficult. Why stay with someone and attempt to hide the friendship behind the current partner’s back? Pretending to be a certified expert at not getting caught? Things just became difficult times three.
Oh, and there’s nothing wrong with a partner being jealous, so long as it’s not the envious, stalking, crazy kind of jealous. Healthy jealousy, meaning they’re the “only one ‘individual’ for me” type, unwilling to share under any circumstances.
On the other hand, if you and your ex share a child or children together, then they are the only matters you two should be discussing, while at the same time reassuring your current partner of your ability to be trusted by including them in “the know.” One thing about a true friend is they stick around for a lifetime, even if you don’t speak with them for awhile or at certain times because they’re understanding. However, your current partner needs to also be understanding. If something unfortunate happened to where your ex is no longer living and you wanted to pay your respects, it shouldn’t be a problem. They should willingly oblige and be at your side, since you willingly obliged by respecting their boundaries.
I have tried to stay friends for the sake of my children however after a while when it’s only you who is keeping in touch and checking up on everyone you get tired after awhile and when you finally drop off the radar and they do not reciprocate you then you know they were not your friend, to begin with.

It depends but It’s not really necessary