Hi,
The shuffling of the cards is completely random and winning is a combo of luck with good cards + skill when planning what card to move.
"Roughly what percentage should be solvable? " - I'm not sure I understand the question.
But to give a rough estimate on how hard this game is. I got the first playable version ready last summer (2023), so a bit over a year ago. I've played this almost daily since and only managed to finish it once.
When I was testing my finishing animations, I had a debug version of only 1 deck (the full game has 2 decks), and then I was able to finish 7/10 games, so it was very easy.