I'm not sure internet native is the best word usage here. I would say it's a race to find the best, most anonymous most transparent and easiest to use currency for online payments.
I reckon the world would slowly turn into one with 2 payment methods, which make up the majority of transactions across the world. One for online purchases, like items from Amazon or buying in game currency for a game and the other one to be used in real life situations.
Anyway. If there was truly a race as described, Bitcoin would be winning it
That's kind of already the case, isn't it? People either use creditcards (Visa/Mastercard), or they use Paypal. Paypal has maybe 5% of the entire online purchasing market.
Bitcoin and other cryptocurrencies are barely a fraction of 1% of online payments. We've actually gone backwards in terms of adoption as a payment in the last few years.