I'm very aware that the game is supposed to be difficult (perhaps the wrong term; there is no difficulty in rolling dice and drawing cards in the correct number and order, only luck), and that even the option to make the game "easier" probably intentionally barely matters - the point of this post was more to define just how unlikely it is, because I (and I'm assuming a few other people on here) am the kind of person to overthink systems like this if I don't actually define them. I definitely appreciate what you're trying to do with this SRD; it's not something that's easy to do outside of a video game format, and outside of play-until-death sandboxes even those usually have somewhat happier endings. I haven't been reading the Twitter conversations, but I would assume that Condition 2 being a complete farce was more of an issue than Condition 1 being unlikely? If that's the case then this post probably can't help fix that anyway (which is obviously the most important one), though it may be useful for gamemakers so they can decide just how unlikely they want victory to be.
Umbra: The simulator code can be found here, but I'd like to restate that this was basically just a rushed way to figure these numbers - it's terrible coding by any standard, cobbled together from half-remembered Java lessons and reused code from a course two years ago. No user-input, any parameter change is accomplished by changing the starting variables for each trial. Deck generation code was outright stolen from one of the first few google results for "java deck code."
Bonus: rolling for escape after every single card (not day) skyrockets the full-random win chance to about 1/20. Starting with the Ace of Hearts raises this to 1/10, and with the Ace of Diamonds as well, about 1/2.
e: Realized my code looks for "King of Aces" so the death condition wasn't properly being checked for. Death odds are probably fairly higher.