the second one means you have to do whether you are willing or

No, it isn't: https://idioms.thefreedictionary.com/be+bound+to

The second example is unnatural. That was bound to happen. (It was predictable or inevitable given the circumstances) Tom got a speeding ticket. It was bound to happen. He always drives over the speed limit. She’s bound to get fired. She’s completely unreliable as an employee.