If you implement proof of stake, for example (where the scarce resource becomes the currency of the network itself, which is a lot trickier to implement and especially quite hard to bootstrap from nothing because in the early days of a currency there's going to be only a few people owning it), then there's no need for the reward process to be a lottery (though it still can be).
The proof of stake proposals don’t seem to work that way. They seem to be more a matter of stakeholders having machines that vote about who is trustworthy.
You need something to be the limiting factor on the amount that you can try to win such a lottery. With proof of work, it's electricity. (assuming everyone is about as efficient as each other) Early on, they talked about it being proportional to cpu-time. That has become more abstract since.
I haven't heard of any cryptos that require you to risk your coins to mine. Sounds like a neat gimmick, but hard to implement.