In over a decade of playing, starting new games many times, and running through the license tests almost every time, I've only been awarded the kiddie prize one time, about three years ago. I had been doing license tests for about an hour and if I recall correctly, on that particular test I had missed bronze by quite a wide margin (about half a second or so, I think), and instead of the standard screen that comes up when you fail the test, I got the kiddie prize instead. I'm pretty sure it was one of the early license tests, maybe for the B license, or one of the very early ones for the A license. It may have been one of tests for I-C, but I doubt it.
Here's some info: My first copy was 1.0, and I played it on an old-style PS1, the kind we all used to call a PSX back in the day. Never got it playing that, even though in the beginning it was just about all I could do just to finish the braking tests. When that copy wore out, I ended up with a 'Greatest Hits' (actual greatest hits, not one of the ones with a sticker on the packaging), for which 1.2 codes work. I was playing this version on a PS2 when I won the kiddie prize, and had failed the test only once before I got the kiddie prize.