'the term implies that people who do have kids are not free and are somehow encumbered'
You took the words out of my mouth @VladmirsPoutine and PPs.
People who have kids are certainly not free and are indeed encumbered. They may like it, but that's their 'burden of blessings' as I heard someone say.
You can't do whatever you like when you have kids. You can't just decide to hop on the next flight to nowhere for the weekend. No such thing as a lie in when you are sick or have a day off work, no quickly nipping anywhere.
I have kids myself and they are my whole life. However, I only really like my own kids. As much as I get on well with kids, and I'll protect and stand up for children, I generally don't want to spend any of my leisure time around other 'small humans'. I don't enjoy pictures of people's kids shoved in my face so that I can say 'Aw'. Who cares?
When I don't have my kids with me, I say I'm 'child-free' for that night or weekend or whatever.
So, yeah, maybe your friend looks at your kids and thinks ' ew, thank goodness I'm child-free!' She has every right to say it whatever way she likes.