Base can't run independent of Ethereum because it finalizes it's transactions on ethereum itself, thus they say, "inhering security of ethereum".
The only way Base would become a direct competitor of BNB, is if the developer team decides to part ways from the Ethereum blockchain (in other words, an independent blockchain or L1). I was only talking about the possibility of this happening in the future. After all, the crypto industry is full of surprises. Whatever Coinbase decides to do with Base in the long run would only be for the best. I think there's a high chance the company will launch a native token for Base, considering that similar projects already did it. Coinbase might be waiting for the next bull run to "pull the trigger".
Using the Base platform on a daily basis could give you the opportunity of earning huge profits after a successful token launch. That's assuming there will be an airdrop. We're just speculating here. No one can predict the future, so we can only hope for the best. Just my thoughts