Here's another kicker: Kangaroo doesn't rely on the birthday paradox. That would be Gaudry-Schost.
Correct.
But in practice, nobody implements original kangaroo with only few kangs. Practical implementation always contains a lot of kangs with random start points and they don't go far, so it's exact or similar to Gaudry-Schost and it does use birthday paradox.
You can make up whatever you want, but without the Kangaroo birthday paradox or any other implementation, they'll just be pretty paperweights.