I think push-ups really call on core strength, as well as shoulder-girdle stability. If either one of those areas is weak, it makes it hard to do on-the-toe push-ups. If you think about it, one exercise used for increasing core strength is the plank, the top position of a push-up. If you can't hold a plank for very long, then you'd have a lot of trouble doing push-ups on the toes.
Some suggestions that might help:
I think that going from bent-knee push-ups to on-the-toes push-ups is a hard transition, because they seem so different. An easier transition is to do on-the-toes push-ups, but from increasingly more difficult angles. The easiest version would be standing and pushing off a wall. Then you'd move to pushing up off the kitchen counter. Then maybe off the edge of the bathtub (if it seems safe) or off the seat of a kitchen chair. Then off a high step. Then a step set at 12". Then a 10" step. And so on, until you are at the floor.
Start at whatever level you can do maybe 5 reps from, for example, and work up to doing 10 reps at this level before moving down to the next level.
You can supplement your 'push-up preparation" program by doing a daily set of planks (start on forearms if necessary, then move to straight arms). Set a timer for 1 minute to begin with, then try to hold the plank for that long. If you can't, then put your knees down momentarily for a short rest, then get back up into plank. Do this as many times as necessary. Once you can hold a plank for a minute, go for a goal of 2 minutes. (If you can already hold a plank for a minute, then add some extra challenge, like lifting up one foot.) On planks, try to hold out as long as you can without arching or bowiing your back. If you start to feel your core 'loosen,' see if you can hold out a bit longer by tightening the area.
