Op, I think you are getting a really hard time here, to be honest!!
I have loads of friends with kids, most of whom are busy and yet are still able and willing to nurture friendships. We all understand those friendships need to be nurtured in a different way now, is all.
You actually sound like you understand, as much as it is ever possible for someone without kids to understand, that parenting is hard work, often relentless, and exhausting. So you've offered to be flexible and pitch in and go with the flow to make your end of the friendship as available as possible. This to me shows you are empathic and care about your friends.
I currently have eight close friends who have a child / children (2 have three kids under 5, 3 have two kids under 3, 1 has three kids under 10, and 2 have 2 kids under 3). All of them I still consider close friends because they have opened their families to me and I have opened my heart to their kids! We go swimming, to the park, we cook together, I cook for them, we visit child friendly museums, I bring wine over, I babysit occasionally. I remember when one had her first, she had PND and her son had awful allergies and reflux and I just used to go take the baby so she could sleep, and in the evenings I would sit with her while she cried and cried and just wanted someone there to watch shit tv with. I have also snuck in and cleaned flats / cooked freezable dinners / etc!
So all this to say, really, LOTS of people have children and still maintain friendships, provided those friends are willing to be flexible and accommodating.
I have lost a friend who I used to be close to after she got pregnant and then gave birth. Always has time for friends with children, never has time for me! Very disappointing actually, and hurtful. I had to re-evaluate and let that go. And that isn't because I am child free and don't get it, but because her attitude changed after she gave birth to one of "I am better than you and my life is soooo different from yours now I have a child". Bollocks.
Interestingly I am now 32 weeks pregnant and she has resurfaced. Sorry love- not interested!