I did have a kid-friendly record player. All you had to do was put the record on the turntable and close the lid. I just had to share with my older sister. I had my Sesame LP's, she had her Partridge Family albums.
I had the first four Sesame albums and a few knockoffs. One was the Peter Pan record with the grotesque drawing of Ernie on the cover. The tracks were covers of tunes from the first season ("Spinning Wheel", "Up Up and Away", etc).
Disneyland records released their version of Sesame's first LP, the most hysterical track is Thurl Ravenscroft (the voice of Tony the Tiger, and " You're a Mean One, Mr. Grinch") singing "Bein' Green" I still can't listen to it and keep a straight face.
The biggest kick my sister and I ever got was playing records at different speeds. All record players had 33 1/3 and 45, but we also loved slowing them down to 16, or speeding them up to 78, where everything sounds like Alvin and the Chipmunks on a sugar rush.
I've found a few old blues records from the 1920's you play at 78, but I've never seen any 16's.