Arimeh, I also wanted to comment.
I also planned a homebirth and ended up transfering to hospital and while it wasn't as extreme as your situation both my son and I also came close to dying.
I completely agree with what others have said, you have not failed at all. An unexpected complication arose that endangered the lives of you and your son and you and your carers acted brilliantly and managed to save both your lives. You did amazingly well.
It is extremely traumatic. It will most likely take you a long time to get over it. I think I'm now most of the way through dealing with my son's birth, 4 years later. But I know I still have some issues from it.
It will of course also have influenced your children and it must have been traumatic for them as well. I wouldn't worry too much about it though. They will probably get over it with time as well and you can help them to understand that it wasn't a normal birth and they shouldn't base their ideas of childbirth on what happened. I don't know but maybe they'll get a chance to witness another birth when they are a bit older. My mother assisted with a home water birth when I was 9 and I went to keep an eye on the labouring woman's other son. We got to come in for the delivery and that memory has really helped me to form a more realistic idea of what natural birth is like and to recognise that mine was not representative for all births.

Good luck and hugs