I have seen many project moved from Ethereum blockchain to BSC because reason of high fee transaction, so Binance maybe has chance to surpass Ethereum based on marketcap and ecosystem. But BNB still hard to do in this bull cycle, maybe BNB will have chance to surpass ETH in another bull cycle, this is according to how next development of both project will run. If ETH 2.0 release and the high cost of transaction fee and network congestion fixed, that mean no room for BNB will surpass ETH.
It's just a matter of time, whether ETH 2.0 will be released soon and resolve existing network congestion and overcome high costs. If Vitalik doesn't finish this soon, maybe as the OP said it could be BSC beyond this. BSC is really interesting once in a year experiencing a tremendous surge once and boosting the BNB coin. Time will tell all, there is no room left to speculate which one is better.