As I mentioned, I only did a small number of runs [about 100]. BTW - I like your title "Retired Expert"!
I think that if you do a large number of runs, you eventually get to about 50%. yes, if you [and the house] are willing to bet micro, nano or pico dollars, then the game can go on forever. But your money is gone, so it is theoretically correct, but not very practical.
Now I am confused. I think I followed the strategy you described yesterday, when you postulated that it would deliver 100% chance of winning. But, now your simulation it gives you a 50% chance of winning.
I was working on the strategy that you describe: Bet half my money [rounded to an integer]; orbet the "ceiling" [twice my original money - Current money].Testing this in a spread sheet [with only about 100 trials] I am winning about 62% of the time.