Is it not feasible to brute force any random private key?

I get the math behind why it's infeasible to brute force any 1 specific private key. It would be like trying to shuffle a pack of cards and get one exact result with all the cards in a specific order.

But something I can't wrap my head around is why you can't just brute force until you get any random private key. It might have access to 0.0001BTC or it might have access to 100BTC it doesn't matter. Surely with modern hardware it's possible to check billions or trillions of combinations per second? Why doesn't anyone do this?

Surely the odds of getting just any random private key in the whole world is far far greater. It's like that birthday probability problem where you only need 28 people in the room to have a >50% chance for 2 people to share a birthday.

Edit: Actually is it even enough to merely guess a private key? Or does this need to go hand in hand with knowing the public key also?