I didn't think "Kyle" was ever the stillborn baby's name, since the K's were for Dr. K, and they didn't even know him before that day. I took the "give him his own name thing" to be "give him a name that's not a K, because he's not a triplet". So while they may have had an idea to name them all with the same letter, "Kyle" was always "Randall", never the stillborn triplet. Just my interpretation - it wasn't especially clear.
My cousin had a stillborn daughter just around this time (I was just old enough that I can remember it). Not sure she got a full funeral, but she most certainly had a name and a burial, because she has a tombstone. But we're Catholic, so... that's maybe different for us.
My parents lost a pair of twins in the early 70s. They were born alive but far too small, and in those days there wasn't much that could be done. They named the weaker one, who only lived a day, after my grandfathers. No one else in the family ever used the names, I think for fear of upsetting my parents, and I know they never wanted that. They would have loved to have had a little "Frankie" running around. I think it would have actually soothed their pain. Instead the only "namesake" my grandpa ever got is a name on a grave. (Not unusual in a lot of families, I know, but in my family we pass down a lot of names. So it's sad to me that his isn't passed down.)