I got it working in Micro-Cap. It works over a nice wide range from 1.5v to 3.3v, which is the range that I happen to care about, and can be made to work at higher voltages by tweaking some of the values. Attached is a simulation showing that it works even when the button presses are 1 second long, which addresses one of the concerns voiced above in this thread.
Regarding the micro-cap simulation, the results are not in complete agreement with the Tina simulation, but there's enough leeway wrt component values that I'm pretty sure it can be made to work on the bench. Unfortunately, I couldn't figure out how to simulate multiple button presses on the same button using micro-cap, so I used two separate time switches to simulate two different presses of the same button. The results are obviously equivalent, even if the resulting micro-cap schematic isn't as pristine as I would have liked.