I'm sorry to hear that this family had to go through this. It sounds as though they were lucky to have such great support with them through all of this. The reality is all those involved in this baby's birth have their own grief to deal with. it's a tough thing as we all go into this work wanting to help babies be born safely and parents to have positive birth experiences. So when things don't go as planned, it's heartbreaking.
Like any loss, it feels so tough to deal with initially, then things get easier with time. You will grieve in your own way. the family will grieve in their own way. As has been said, as doulas and midwives, we often have to be the supportive ones in these difficult times. I think this helps us in our grief - to be there with and for the family - as it's something we can do for them. it gives us the opportunity to share some of the common grief between us - to remember this little one.
I think it's common for our usual supports (friends and family) may not 'get' why we're so upset and so it puts us in a funny place of not being able to fully rely on our usual supports when these things happen. I agree that it's important to do what you can for yourself. be good to yourself, allow yourself to cry when you need to and also allow yourself to laugh when you want to. Know that those of us who have been through similar situations understand. It's a really tough job that you took on - supporting a family dealing with such grief as they birthed their baby. The fact that you were there with them, supported them, shared what joy existed in the moment, speaks volumes to your capacity as a doula. don't underestimate the intensity of what you went through. This will surely have strengthened you as a doula.
I know what you mean about not sure how to process this. I really can't give you suggestions more than this. I was a midwife for a family in the Philippines a few years ago when their baby girl was born stillborn. Long story, but it all happened pretty quickly. So suddenly I find myself in a situation where I have to be a support to the family, and I washed and dressed this little one for her burial, at the family's request (something that I never thought I'd never have to do). Part of me wanted to just break down and cry rather than deal with what was going on. But, my job was the be the midwife, so I buried those emotions as best I could and got on with the work. I got through the day. I think I buried those emotions so deep that they had a hard time coming out. Like you, I felt like I should be upset, but not sure how to deal with all of it. I found myself crying over silly things totally unrelated to this family - the tears had to come out one way or another. Visiting with the family helped me with my grief. I found myself thinking about this birth every so often even once I was back home and eventually I could be at peace with my emotions but it took time.
I don't know if that helps. Hugs to you as you deal with all of this.