This is so hard, my heart goes out to you.
BOTH parents loved each other to have kids. It's a good thing to remember that and to support your partner. That will show your children that even though you aren't married any longer, you'll always be a supportive family.
It doesn't matter who did what or feels love for someone else.
Tell your kids you love them and they will spend time with both of you. Tell them that you are still parents together, even if you aren't living together.
Remind them that there are many definitions of 'family'.
The best advice I can give (btdt) is to forgive your partner and forgive yourself. Kids take their cues from you. You've made a mistake (or quite a few) and this is how you move on after a mistake. You forgive, apologize, and try to make it work for everyone.
I still love my ex -- just enough. I want him healthy and happy and to be a good parent, so I support him in those areas. We do our child's birthday together in a restaurant. We live close enough that our kid can be bussed to either home but from their only school. We take turns doing holidays. I have a weekend and a school week. He then has the same. We text each other about our daughter and call if it is more complicated.
He had an affair and I was pissed as hell. I can do nothing to change it and as far as our daughter knows -- all that happened was that her parents failed at their marriage. She is old enough to know all about it, but unless someone else has told her the gory details, they were kept from her. We reassured her that she had nothing to do with our marriage break up and that is the truth.
She has tried to pit one of us against the other to get her way, but we always just text to make sure that Dad/Mum did say she could do whatever it is.
Now that you edited:
My fear is that simply stating that the parents do not love each other any more will not be the truth and will put exceedingly the blame on the loving partner. This is a crucial issue as the non-loving partner is facing a personal crisis and partly rejecting the parenting role.
A parent who is partially rejecting their parental role is not acting in the best interests of their children. These aren't an 'unsuccessful' experiment. There is no return policy on having one child let alone two.
So, I'd try hard to just love the children, encourage them and step up your game. It will be harder being a single parent, especially for children suffering a loss.
Please try not to explain or denegrade your ex. Of course you are angry, but the kids will suffer more if they see you angry. So it's fine to tell them that you are sorry it happened. It's fine to agree that you wish their parent was around more often and doing with more with them, but it is not fine to say that the missing parent is 'bad' or 'wrong'. Let the children ask for themselves and tell them that you can't answer or that you are unable to explain it.
The children will be angry with you if you make your 'ex' into a bad person. They will decide that for themselves.
Family counselling (Kids plus you and with ex if possible) with an impartial professional is great.
As I said, this is so hard and you have to do more at a time that you are not only grieving, but also doing more of the work. However, this is what good parents do.
Please keep us posted on how you are doing. We all have had experiences that may help you on your journey through this one.
Best of luck.