

This is cute but not practical.
Memorising all 2^n for 0≥n≥10 (1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024) is reasonable, but you’ll need to add a lot of them at the same time to convert the counting in your hands to a base 10 number (that’ll use elsewhere). Stuff like 256+128+64+8+2+1; it isn’t difficult but laborious, you know?
Plus the gestures can be sometimes awkward, depending on how flexible your hand is. For example, at least for me it’s a bit tricky to lift the ring finger up without either the middle or the pink.
I have a different strategy to count large numbers by hand. It’s up to 12 with one hand, 144 with both. But it feels comfortable, and rather intuitive:

Put the tip of your thumb on the indicated places to count 1, 2, 3… 12. With one hand; if you want to count past 12, use the other hand to count dozens (12, 24, 36… 144).
Provided you memorised the multiplication table for 12, for any given number you’ll perform at most a single addition, like 84+7 or similar.
(I have a suspicion the Sumerians counted this way with one hand, and one finger per dozen with the other. That’s why a lot of their units 5*12=60 as a basis.)






Worst hypothesis, even if counting just to 8, you’re still using a single hand, so it’s an improvement over counting to 5 with fingers. Couple it with the other hand and you’re counting up to 64.
There’s also the Chisanbop method @hesh@quokk.au mentioned, for that your unbending joint shouldn’t be an issue, and it allows you to count to 9 with a single hand (99 with both).