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
At my first thoughts about reading this topic, the term native for me means, a cryptocurrency suitable for people without an internet connection. But upon look at the actual website, the race focuses on the cryptocurrency itself. If the OP just provides a bit of an introductory sentence, it would be better for us to easily understand.
Going back to the topic, the race for the cryptocurrency might still depend on what aspect of a transaction will be the goal. If they will consider the transaction speed, bitcoin will not be the best. As well as considering the transaction fee, there is surely some cryptocurrency that is much more suitable for this such as XRP and ETH. If bitcoin will utilize the lightning network then bitcoin will take lead for majority of the cryptocurrency.