Is Bitcoin's safety is only in high number of possible private keys or there is something else that I'm missing or don't know.

Since there is a connection between Private Key and Public Key it makes things like fishing for addresses possible. Like generate a random private key and generate public key for that private key, check if the balance is non zero and use private key to get BTC from there. I know there are like really really high number of possible private keys and someone finding my private keys is a low chance. But it still feels vulnerable to have the shield of just random luck. I feel like if there are enough people with enough computer power is trying this method someone's money would be reached. If someone manages to find my private keys this way that it doesn't really matter if it is a low probability or not.

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I always thought it would better to have public keys and private keys unrelated and should need to have them both kind of a USERNAME/PASSWORD kind of couple. But as far as I know this is not the case. Can someone explain me if I'm missing something?